Improved bounds for five-term arithmetic progressions

Abstract

Let $r_5(N)$ be the largest cardinality of a set in $\{1,\ldots,N\}$ which does not contain 5 elements in arithmetic progression. Then there exists a constant $c\in (0,1)$ such that

\[r_5(N)\ll \frac{N}{\exp\!((\!\log\log N)^{c})}.\]

Our work is a consequence of recent improved bounds on the $U^4$-inverse theorem of J. Leng and the fact that 3-step nilsequences may be approximated by locally cubic functions on shifted Bohr sets. This, combined with the density increment strategy of Heath–Brown and Szemerédi, codified by Green and Tao, gives the desired result.

1. Introduction

Let $[N] = \{1,\ldots, N\}$ and $r_k(N)$ denote the size of the largest $S\subseteq\{1,\ldots, N\}$ such that S has no k-term arithmetic progressions. The first nontrivial upper bound on $r_3(N)$ came from work of Roth [ 28 ] which proved

\[r_3(N)\ll N(\!\log\log N)^{-1}.\]

A long series of works improved the bounds in the case $k=3$ . This includes works of Heath–Brown [ 16 ], Szemerédi [ 35 ], Bourgain [ 5, 6 ], Sanders [ 30, 31 ], Bloom [ 2 ], and Bloom and Sisask [ 3 ]. Recently, in breakthrough work, Kelley and Meka [ 18 ] proved that

\[r_3(N)\ll N\exp\!(\!-\!\Omega((\!\log N)^{1/12}));\]

modulo the constant $1/12$ this matches the lower bound construction of Behrend [ 1 ]. The constant $1/12$ was refined to $1/9$ in work of Bloom and Sisask [ 4 ].

For higher k, establishing that $r_k(N) = o(N)$ was a long standing conjecture of Erdös and Turán. In seminal works, Szemerédi [ 33 ] first established the estimate $r_4(N) = o(N)$ and then in later work Szemerédi [ 34 ] established his eponymous theorem that

\[r_k(N) = o(N).\]

Due to the uses of van der Waerden theorem and the regularity lemma (which was introduced in this work), Szemerédi’s estimate was exceedingly weak. In seminal work Gowers [ 7, 8 ] introduced higher order Fourier analysis and proved the first “reasonable” bounds for Szemerédi’s theorem:

\[r_k(N) \ll N(\!\log\log N)^{-c_k}.\]

The only previous improvement to this result for $k\ge 4$ was work on the case $k=4$ of Green and Tao [ 13, 14 ] which ultimately established that

\[r_4(N)\ll N(\!\log N)^{-c}.\]

Our main result is a “quasi-logarithmic” bound in the case $k = 5$ .

Theorem 1·1. There is $c\in(0,1)$ such that

\[r_5(N)\ll\frac{N}{\exp\!((\!\log\log N)^{c})}.\]

Remark. Throughout this paper, we will abusively write $\log$ for $\max(\!\log\!(\cdot), e^e)$ . This is to avoid trivial issues regarding inputs which are small.

Our work relies crucially on a recent improved inverse theorem for the Gowers $U^4$ -norm due to the first author [ 23 , theorem 7]. We first formally define the Gowers $U^k$ -norm.

Definition 1·2. Given $f\colon\mathbb{Z}/N\mathbb{Z}\to\mathbb{C}$ and $k\ge 1$ , we define

\[\lVert f\rVert_{U^k}^{2^k}=\mathbb{E}_{x,h_1,\ldots,h_k\in\mathbb{Z}/N\mathbb{Z}}\Delta_{h_1,\ldots,h_k}f(x)\]

where $\Delta_hf(x)=f(x)\overline{f(x+h)}$ is the multiplicative discrete derivative (extended to vectors h in the natural way). This is known to be well-defined and a norm (seminorm if $k=1$ ).

Theorem 1·3. There exists an absolute constant $C\ge 1$ such that the following holds. Let N be prime, let $\delta \gt 0$ , and suppose that $f\colon\mathbb{Z}/N\mathbb{Z}\to\mathbb{C}$ is a 1-bounded function with

\[\lVert f\rVert_{U^4(\mathbb{Z}/N\mathbb{Z})}\ge\delta.\]

There exists a degree 3 nilsequence $F(g(n)\Gamma)$ such that it has dimension bounded by $(\!\log\!(1/\delta))^C$ , complexity bounded by C, such that F is 1-Lipschitz, and such that

\[\big|\mathbb{E}_{n\in[N]} f(n)\overline{F(g(n)\Gamma)}\big|\ge1/\exp\!((\!\log\!(1/\delta))^{C}).\]

A key manoeuvre in this paper is our deduction of a variant of the $U^4$ -inverse theorem which lends itself to the analysis of multiple nilsequences simultaneously and may be of independent interest. Although it is known that having a large $U^4$ -norm does not necessarily imply direct correlation with a cubic phase function due to the existence of bracket polynomials, one can hope to achieve correlation on a dense, structured host set (for us, a Bohr set).

Proposition 1·4. There exists an absolute constant $C\ge 1$ such that the following holds. Let N be prime, let $\delta \gt 0$ , and suppose that $f\colon\mathbb{Z}/N\mathbb{Z}\to\mathbb{C}$ is a 1-bounded function with

\[\lVert f\rVert_{U^4(\mathbb{Z}/N\mathbb{Z})}\ge\delta.\]

There exist $S\subseteq (1/N)\mathbb{Z}$ with $|S|\ll (\!\log\!(1/\delta))^C$ , and $y\in\mathbb{Z}/N\mathbb{Z}$ such that the following holds: there is a locally cubic phase function $\phi\colon y+B(S,1/100)\to\mathbb{R}/\mathbb{Z}$ such that

\[|\mathbb{E}_{t\in\mathbb{Z}/N\mathbb{Z}}\mathbb{1}_{t\in y + B(S,1/100)}f(t)e(\!-\!\phi(t))|\gg1/\exp\!((\!\log\!(1/\delta))^C).\]

We refer the reader to Definitions 2·1 and 2·2 for precise definitions of Bohr set and locally cubic phase function. We remark that an analogous result to Proposition 2·3 for higher $U^k$ -norms is false; this can be seen by examining the function $e(\{an\{bn\}\}\{cn\} dn)$ . We discuss this issue in more detail in Section 1·1.

1·1. Proof outline

The starting point of our work involves combining the density increment strategy of Heath-Brown [ 16 ] and Szemerédi [ 35 ], which was reformulated in a robust manner by Green and Tao [ 13 ] when proving that $r_4(N)\ll N\exp\!(\!-\!\Omega(\sqrt{\log\log N}))$ , with the improved $U^4$ -inverse theorem Theorem 1·3 of the first author. The crucial difference between the density increment strategy of Roth [ 28 ] versus Heath-Brown [ 16 ] and Szemerédi [ 35 ] is that one finds correlations with “many functions” to deduce a density increment.

If we apply Theorem 1·3 directly with the density increment strategy as codified by Green and Tao [ 13 ], we would at the very least need that, given a polynomial sequence g(n) with $g(0) = \operatorname{id}_{G}$ on a nilmanifold $G/\Gamma$ of degree 3 with complexity M and dimension d, we have

\[\min_{1\le n\le N}d_{G/\Gamma}(\operatorname{id}_G,g(n))\ll M^{O(d^{O(1)})}N^{-1/d^{O(1)}}.\]

When g(n) is a polynomial sequence of degree 3 on the torus such results can be derived from work of Schmidt [ 32 ] on small fractional parts of polynomials. While directly deriving such a result for nonabelian nilmanifolds does not appear implausible, at present the distribution theory of nilmanifolds only has such “polynomial in d” dependencies when dealing with test functions having vertical frequency, due to work of the first author [ 22 ].

The crucial step in our work therefore is solving such a Schmidt-type problem for nilmanifolds via representing 3-step nilmanifolds as sums of exponentials of locally cubic functions on Bohr sets (see Proposition 2·3). The analogue of such a result for 2-step nilmanifolds (without bounds) appears in work of Green and Tao [ 10 , proposition 2·3]. That such a result is true is a miracle of small numbers which is most easily seen via bracket polynomials. (In work of Green and Tao [ 13 ] regarding $r_4(N)$ , one operates with a version of the $U^3$ -inverse theorem proven directly for locally quadratic functions on Bohr sets.)

At an informal level, the $U^4$ -inverse theorem may be rephrased as follows: if $\lVert f\rVert_{U^4}$ is large for 1-bounded f, then f correlates with a bracket exponential phase e(H(n)) where H(n) is (essentially) a sum of terms of the form

\[an^3, an^2\{bn\}, an\{bn^2\}, an\{bn\}\{cn\}, an\{\{bn\}\{cn\}\}, \{an\}\{bn\}\{cn\}, an\{bn\{cn\}\} \;\mathrm{mod}\;1\]

plus terms which are obviously of “lower degree”. By the work of Green and Tao, various lower order terms may be viewed as quadratic functions on Bohr sets. Now note that

\[\{x\}y + y\{x\} = xy - \lfloor x\rfloor \lfloor y\rfloor + \{x\}\{y\}.\]

Therefore

\[\{x\}y + y\{x\} = xy + \{x\}\{y\} \;\mathrm{mod}\;1.\]

Furthermore note that $e(\{x\}\{y\})$ is, after appropriate smoothing, a Lipschitz function on $(\mathbb{R}/\mathbb{Z})^2$ and therefore is well-approximated by a weighted sum of exponentials of the form $e(kx + \ell y)$ for $k,\ell\in \mathbb{Z}$ . Given this (and noting the analogous fact for $e(\{x\}\{y\}\{z\})$ ) we may rewrite our basis of degree 3 functions as

\[an^3, an^2\{bn\}, an\{bn\}\{cn\} \;\mathrm{mod}\;1;\]

the most crucial of these manipulations is

\[an\{bn\{cn\}\} = abn^2\{cn\} - \{an\}bn\{cn\} \;\mathrm{mod}\;1.\]

The miracle is that we do not have any nested $\{\cdot\}$ and all of the brackets surround linear functions. Therefore, upon restricting the fractional parts to lie in certain narrow ranges away from the discontinuities in the fraction part (i.e., Bohr set-type conditions) the functions $an^2\{bn\}, an\{bn\}\{cn\}$ are seen to be “locally cubic”. That is, discrete fourth-order derivatives vanish given that all points in the corresponding 4-dimensional hypercube lie in an appropriate Bohr set. The existence of such a miracle can be seen by examining carefully all “fractional part” expressions required in work of Green–Tao–Ziegler [ 12 , appendix E] on the $U^4$ -inverse theorem. For the $U^5$ -inverse theorem and higher, we have the function $e(\{an\{bn\}\}\{cn\} dn)$ and fractional part identities do not allow one to “remove iterated floor functions”.

To actually prove the desired representation of a step 3 nilsequence, we partition the nilmanifold via a partition of unity. Operating within a partition of unity, we may manipulate the nilmanifold (as in [ 10 , proposition 2·3]) via explicitly operating in Mal’cev coordinates of the first kind. This allows us to manipulate the floor and fractional expressions as suggested by the bracket polynomial formalism in the previous paragraphs. Identifying various fundamental domains with the torus and applying Fourier expansion appropriately eventually gives the desired decomposition.

We remark for technical reasons it turns out to be useful to manipulate our nilsequence to be N-periodic and living on a nilmanifold for which the Lie bracket structure constants are integers which are sufficiently divisible. The first task is accomplished via appropriately quantifying a construction of Manners [ 24 ] (see Proposition C·2) and the second is accomplished via a “clearing denominators” trick on the Mal’cev basis (see Lemma C·1). One can see that such a manipulation might be useful by noting that if the structure constants of $G/\Gamma$ are sufficiently divisible then $\psi_{\exp}(\Gamma) = \mathbb{Z}^d$ where $\psi_{\exp}$ is the map to Mal’cev coordinates of the first kind (see Lemma 2·5). In general $\psi_{\exp}(\Gamma)$ is not even a lattice in $\mathbb{Q}^d$ .

Finally, given a correlation with a locally cubic function on a Bohr set, we are now in position to run the density increment strategy of Heath-Brown [ 16 ] and Szemerédi [ 35 ] as codified by Green and Tao [ 13 ]. Our proof is very close to that of Green and Tao [ 13 ], although there are slight simplifications afforded in our situation since the dimension of our Bohr sets are “quasi-logarithmic”. In particular, it is possible to operate by considering single atoms in the Bohr partition and avoid the use of regular Bohr sets. Our situation at this point is closely analogous to having access to the $U^3$ -inverse theorem of Sanders [ 29 ] and aiming to prove bounds of the form $r_4(N)\ll N \exp\!(\!-\!(\!\log\log N)^c)$ .

Notation. We use standard asymptotic notation. Given functions $f=f(n)$ and $g=g(n)$ , we write $f=O(g)$ , $f \ll g$ , $g=\Omega(f)$ , or $g\gg f$ to mean that there is a constant C such that $|f(n)|\le Cg(n)$ for sufficiently large n. We write $f\asymp g$ or $f=\Theta(g)$ to mean that $f\ll g$ and $g\ll f$ , and write $f=o(g)$ or $g=\omega(f)$ to mean $f(n)/g(n)\to0$ as $n\to\infty$ . Subscripts on asymptotic notation indicate dependence of the bounds on those parameters. We will use the notation $[x] = \{1,2\ldots,\lfloor x\rfloor\}$ .

We use the notations of Appendix A with regards to nilmanifolds. Write $\Delta_hf(x)=f(x)\overline{f(x+h)}$ for the multiplicative discete derivative and $\partial_hf(x)=f(x)-f(x+h)$ for the additive discrete derivative (for functions over appropriate domains and codomains). The Gowers $U^s$ -norm on a finite abelian group G is then defined via

\[\lVert f\rVert_{U^s}^{2^s}\;:\!=\;\mathbb{E}_{x,h_1,\ldots,h_s\in G}\Delta_{h_1,\ldots,h_s}f(x)\]

for $f\colon G\to\mathbb{C}$ , which is known to be well-defined and a norm for $s\ge 2$ and a seminorm for $s=1$ .

1·2. Organisation of paper

In Section 2 we prove the main technical result regarding approximating nilsequences as local cubics on Bohr sets. In Section 3 we deduce the main result via a density increment argument. In Appendix A we collect various conventions and definitions regarding nilmanifolds. In Appendix B we essentially reproduce [ 13 , appendix A] and note that it verbatim extends to higher degree polynomials. In Appendix C, we manipulate Theorem 1·3 to prove Theorem C·3 which gives correlation with a periodic nilsequence where the underlying nilmanifold has appropriately divisible Lie bracket structure constants. Finally, in Appendix D we collect a number of deferred technical lemmas.

2. Converting nilsequences to local cubics on a Bohr set

The key idea is to work with a presentation of our nilsequence coming from Theorem 1·3, and manipulate it into an approximate form composed of locally cubic functions on Bohr sets.

We will first define Bohr sets formally.

Definition 2·1. Given $S\subseteq\mathbb{Z}/N\mathbb{Z}$ and $\rho\in(0,1)$ , we define the (centered) Bohr set

\[B(S,\rho)\;:\!=\;\{x\in\mathbb{Z}/N\mathbb{Z}\colon\lVert\xi x/N\rVert_{\mathbb{R}/\mathbb{Z}} \lt \rho\text{ for all }\xi\in S\}.\]

Given $\alpha=(\alpha_\xi)_{\xi\in S}\in(\mathbb{R}/\mathbb{Z})^S$ , we define the uncentered Bohr set

\[B_\alpha(S,\rho)\;:\!=\;\{x\in\mathbb{Z}/N\mathbb{Z}\colon\lVert\xi x/N-a_\xi\rVert_{\mathbb{R}/\mathbb{Z}} \lt \rho\text{ for all }\xi\in S\}.\]

The parameter $|S|$ is the rank and $\rho$ is the radius.

We next define the notion of local polynomial structure; we will care specifically about the case of degree $s=3$ .

Definition 2·2. Given $S\subseteq G$ and $f\colon S\to H$ , we say that f is locally degree $s\ge0$ if for all $x,h_1,\ldots,h_{s+1}$ such that $x+\sum_{j=1}^{s+1}\epsilon_jh_j\in S$ for all $\epsilon=(\epsilon_1,\ldots,\epsilon_{s+1})\in\{0,1\}^{s+1}$ , we have

\[\partial_{h_1,\ldots,h_{s+1}}f(x)=0.\]

Remark. We will primarily be concerned with $S\subseteq\mathbb{Z}/N\mathbb{Z}$ and $H = \mathbb{R}/\mathbb{Z}$ .

Proposition 2·3. Suppose we are given a degree 3 nilmanifold $G/\Gamma$ of dimension d, complexity M, and all Lie bracket structure constants divisible by 12. Furthermore suppose that $F\colon G/\Gamma\to\mathbb{C}$ is smooth with Lipschitz norm bounded by 1 and let $\eta\in(0,1/2)$ .

Then there exist data such that

\[\sup_{n\in\mathbb{Z}/N\mathbb{Z}}\bigg|F(g(n)\Gamma) - \sum_{i\in I}w_i\mathbb{1}_{n\in y_i + B(S,1/100)}e(\phi_i(n))\bigg|\le \eta\]

such that:

1. (i) $S\subseteq (1/N)\mathbb{Z}$ with size at most $d+1$ ;

2. (ii) I is an index set of size at most $O(M \eta^{-1})^{O(d^{O(1)})}$ and $|w_i|\le O(M \eta^{-1})^{O(d^{O(1)})}$ for all $i\in I$ ;

3. (iii) $y_i\in\mathbb{Z}/N\mathbb{Z}$ for all $i\in I$ ;

4. (iv) $\phi_i\colon\mathbb{Z}/N\mathbb{Z}\to\mathbb{R}/\mathbb{Z}$ is locally cubic on $y_i + B(S,1/100)$ .

Then, Proposition 1·4 directly follows from a slightly modified version of Theorem 1·3 (namely Theorem C·3) and Proposition 2·3. Theorem C·3 allows one to essentially assume that the underlying nilsequence is N-periodic and various structure constants are sufficiently divisible.

Proof of Proposition 1·4. By Theorem C·3, there exists a nilmanifold $G/\Gamma$ with function F and a polynomial sequence g(n) such that

\[\big|\mathbb{E}_{n\in \mathbb{Z}/N\mathbb{Z}}f(n)\overline{F(g(n)\Gamma)}\big|\ge 1/\exp\!((\!\log\!(1/\delta))^C) =: 2\eta\]

with $g(0) = \operatorname{id}_G$ and $G/\Gamma$ having dimension d, complexity at most M and all structure constants divisible by 12 with

\begin{align*}d&\le (\!\log\!(1/\delta))^{O(1)}\\M&\le \exp\!((\!\log\!(1/\delta))^{O(1)}).\end{align*}

We now approximate $F(g(n)\Gamma)$ by a function H(n) such that $\sup_{n\in \mathbb{Z}/N\mathbb{Z}}|F(g(n)\Gamma)-H(n)|\le\eta$ using Proposition 2·3. As f is 1-bounded, we immediately have

\[\big|\mathbb{E}_{n\in\mathbb{Z}/N\mathbb{Z}}f(n)\overline{H(n)}\big|\ge \big|\mathbb{E}_{n\in\mathbb{Z}/N\mathbb{Z}}f(n)\overline{F(g(n)\Gamma)}\big| - \mathbb{E}_{n\in\mathbb{Z}/N\mathbb{Z}}|H(n)-F(g(n)\Gamma)|\ge \eta.\]

We have some additional guarantees on the structure of H (in particular, some associated data $S,I,w_i,y_i,\phi_i$ ). Applying Pigeonhole on the index set I and using that the $w_i$ are sufficiently bounded, there exist $y\in\mathbb{Z}/N\mathbb{Z}$ , S a set of at most size $d+1$ , $\rho=1/100$ , and $\phi\colon\mathbb{Z}/N\mathbb{Z}\to\mathbb{R}/\mathbb{Z}$ locally cubic on $y + B(S,\rho)$ such that

\[\big|\mathbb{E}_{n\in\mathbb{Z}/N\mathbb{Z}}f(n) \mathbb{1}_{n\in y+ B(S,\rho)}e(\!-\!\phi(n))\big|\ge 1/\exp\!((\!\log\!(1/\delta))^{O(1)}).\]

In order to prove Proposition 2·3, we will need a number of quantitative and structural lemmas about nilsequences. These arguments are deferred to Appendix D. We first require the following quantitative partition of unity for nilmanifolds.

Lemma 2·4. Fix $\varepsilon\in (0,1/2)$ and a nilmanifold $G/\Gamma$ of degree s, dimension d, and complexity M. There exists an index set I and a collection of nonnegative smooth functions $\tau_j\colon G/\Gamma\to\mathbb{R}$ for $j\in I$ such that:

1. (i) $L\;:\!=\; (M/\varepsilon)^{O_s(d^{O_s(1)})}$ ;

2. (ii) for all $g\in G$ , we have $\sum_{j\in I}\tau_j(g\Gamma) = 1$ ;

3. (iii) $|I|\le L$ ;

4. (vi) for each $j\in I$ , there exists $\beta\in [\!-\!L,L]^{d}$ so that for any $g\Gamma \in \operatorname{supp}(\tau_j)$ there exist $g'\in g\Gamma$ such that $\log_G(g')\in \prod_{i=1}^d[\beta_i-\varepsilon,\beta_i+\varepsilon]$ ;

5. (v) $\tau_j$ are L-Lipschitz on $G/\Gamma$ .

We also require the algebraic fact that if the Lie bracket structure constants of $G/\Gamma$ are sufficiently divisible then $\psi_{\exp}(\Gamma) = \mathbb{Z}^d$ .

Lemma 2·5. There exists an integer $C_s\ge 1$ such that the following holds. Fix a nilmanifold $G/\Gamma$ of degree s and a Mal’cev basis $\mathcal{X}$ such that all Lie bracket structure constants of $G/\Gamma$ are divisible by $C_s$ . Then $\psi_{\exp}(\Gamma) = \mathbb{Z}^d$ .

Remark. We may take $C_3 = 12$ .

Next we need the conversion between the nilmanifold distance and distance in coordinates of the first kind (on $\mathbb{R}^d$ ).

Lemma 2·6. Fix a nilmanifold $G/\Gamma$ of degree s, dimension d, and complexity M. If $\lVert\psi_{\exp}(x) - \psi_{\exp}(y)\rVert_\infty\le\varepsilon$ and $\lVert\psi_{\exp}(x)\rVert_\infty\le L$ then

\[d_{G/\Gamma}(x\Gamma,y\Gamma)\le\varepsilon(LM)^{O_s(d^{O_s(1)})}.\]

We also require the following basic estimate regarding Fourier expansion of Lipschitz functions on the torus; this follows immediately by quantifying the proof in [ 36 , proposition 1·1·13] (see e.g. [ 26 , lemma A·8]).

Lemma 2·7. Fix $0 \lt \varepsilon \lt 1/2$ , and let $F\colon(\mathbb{R}/\mathbb{Z})^d\to\mathbb{C}$ with $\lVert F\rVert_{\mathrm{Lip}}\le L$ with metric $d(x,y) = \max_{1\le i\le d}\lVert x_i-y_i\rVert_{\mathbb{R}/\mathbb{Z}}$ for $x,y\in (\mathbb{R}/\mathbb{Z})^d$ . There exists an absolute constant $C \gt 0$ such that there exist $c_{\xi}$ with $\sum_{\xi}|c_{\xi}|\le (3CLd\varepsilon^{-1})^{5d}$ and

\[\sup_{x\in (\mathbb{R}/\mathbb{Z})^d}\bigg|F(x) - \sum_{|\xi|\le (CLd\varepsilon^{-1})^{2}}c_{\xi}e(\xi\cdot x)\bigg|\le \varepsilon.\]

We finally require the following elementary lemma regarding how local degree acts with respect to multiplication; this is once again deferred to Appendix D. (Recall we are defining local degree with respect to $\partial$ as opposed to $\Delta$ , and the following cannot be altered to only require that the functions reduced $\mathrm{mod}\;1$ have local degree.)

Lemma 2·8. If $S\subseteq G$ and $f_j\colon S\to\mathbb{C}$ are locally degree $d_j$ functions, then $g=\prod_{j=1}^kf_k$ is a locally degree $\sum_{j=1}^kd_j$ function.

We are ready to proceed.

Proof of Proposition 2·3. We break the proof into a series of steps.

Step 1. Polynomial sequence and group multiplication in coordinates. Note that as we are dealing with nilpotent groups of step at most 3 we have

\[\log\!(e^Xe^Y)=X+Y+\frac{1}{2}[X,Y]+\frac{1}{12}([X,[X,Y]]-[Y,[X,Y]])\]

by the Baker–Campbell–Hausdorff formula. Let $d_j=\dim G_{1}-\dim G_{j+1}$ and let our Mal’cev basis be $(X_k)_{k\in[d]}$ such that $(X_k)_{k \gt d_j}$ spans $\log G_{j+1}$ for $j\in\{0,1,2\}$ . Note that by definition $\Gamma$ is the set of elements of the form $\prod_{i=1}^{d}\exp\!(t_iX_i)$ for $t_i\in\mathbb{Z}$ . Recall that we have

\[[X_i,X_j]=\sum_{k=1}^dc_{ijk}X_k\]

for $c_{ijk}$ integers divisible by 12 and of absolute value at most M due to the complexity assumption. As $[\log G_i, \log G_j]\subseteq \log G_{i+j}$ , we have that $c_{ijk}=0$ for $k\le d_{a+b+1}$ if $i \gt d_a$ and $j \gt d_b$ .

As g(n) is a polynomial sequence with $g(0) = \operatorname{id}_G$ , by Taylor expansion we have

\[g(n) = g_1^{n}g_2^{\binom{n}{2}}g_3^{\binom{n}{3}}\]

with $g_i\in G_i$ . Therefore

\begin{align*}\log\!(g(n)) &= \log\!\bigg(\!\exp\!(n\log\!(g_1))\exp\bigg(\binom{n}{2}\log\!(g_2)\bigg)\exp\bigg(\binom{n}{3}\log\!(g_3)\bigg)\bigg)\\&=n\log\!(g_1) + \binom{n}{2}\log\!(g_2) + \frac{n}{2}\binom{n}{2}[\log\!(g_1),\log\!(g_2)] + \binom{n}{3}\log\!(g_3)\end{align*}

with $[\log\!(g_1),\log\!(g_2)]\in\log\!(G_3)$ . Therefore we have

(2·1) \begin{equation}g(n)=\exp\!\bigg(\sum_{j=1}^{d}p_j(n)X_j\bigg)\end{equation}

with $p_j(0) = 0$ and $p_j$ being at most degree 1 if $1\le j\le d_1$ , at most degree 2 if $d_1+1\le j\le d_2$ , and at most degree 3 if $d_2+1\le j\le d_3=d$ . We next realise multiplication of two elements corresponding to Mal’cev coordinates of the first kind as

\begin{align*}&(x_1,\ldots,x_d)\ast(y_1,\ldots,y_d)\\&=(x_1+y_1,\ldots,x_{d_1}+y_{d_1},x_{d_1+1}+y_{d_1+1}+\phi_{d_1+1}(x_{\le d_1},y_{\le d_1}),\ldots,x_{d_2}+y_{d_2}\\&+\phi_{d_2}(x_{\le d_1},y_{\le d_1}), x_{d_2+1}+y_{d_2+1}+\phi_{d_2+1}(x_{\le d_2},y_{\le d_2})+\varphi_{d_2+1}(x_{\le d_1},y_{\le d_1}),\ldots,x_d+y_d\\&+\phi_d(x_{\le d},y_{\le d})+\varphi_d(x_{\le d_1},y_{\le d_1}))\end{align*}

where $\phi_j$ are bilinear forms with integral coefficients and $\varphi_j$ are cubic forms with integral coefficients. This is an immediate consequence of the Baker–Campbell–Hausdorff formula and using the assumption that the Lie bracket structure constants are all divisible by 12. Furthermore these coefficients are of height at most $O(Md)^{O(1)}$ and since $[G_2,G_2]\subseteq G_4=\operatorname{Id}_G$ , the bilinear form $\phi_j$ for $j\in[d_1+1,d_2]$ has all coordinates 0 on the box $[d_1+1,d_2]^2$ .

Step 2. Explicit representation of nilsequence with coordinates. We can represent the coordinates of $g(n)\Gamma$ in a fundamental domain with respect to Mal’cev coordinates of the first kind via iterated floor and fractional parts. These coordinates are not smooth at the boundary and thus we decompose F via a partition of unity (and using different fundamental domains for each part). This will allow us to manipulate such coordinate functions without needing to worry very precisely about the minor discontinuities.

Let $L = O(M)^{O(d^{O(1)})}$ . We write

\[F=\sum_{i\in I}F_i\]

with $F_i=F\tau_i$ where $\tau_i$ is as in Lemma 2·4 applied with parameter $\varepsilon=10^{-3}$ . This will allow us to represent $F\tau_i$ on the fundamental domain (with respect to Mal’cev coordinates of the first kind) $\prod_{j=1}^d[\beta_j^{(i)}-1/2,\beta_j^{(i)}+1/2)$ , where $\beta^{(i)}\in [\!-\!L,L]^{d}$ and

\[\operatorname{supp}(F\tau_i)\subseteq\operatorname{supp}(\tau_i)\subseteq\prod_{j=1}^d[\beta_j^{(i)}-10^{-3},\beta_j^{(i)}+10^{-3}).\]

We now define nonstandard (shifted) floor and fractional part functions for each $i\in I$ and $j\in[d]$ so that

\begin{align*}\{t\}_{i,j} &\equiv t \;\mathrm{mod}\;1\\t &= \{t\}_{i,j} + \lfloor t\rfloor_{i,j}\\\{t\}_{i,j}&\in [\beta_j^{(i)}-1/2,\beta_j^{(i)}+1/2).\end{align*}

For nearly the entire remainder of the proof, we will focus on massaging the representation of $F_i(x)$ into a more convenient form. Given $x\in G$ such that $\psi_{\exp}(x)= (x_1,\ldots,x_d)$ , consider $\gamma$ with Mal’cev coordinates of the first kind:

(2·2) \begin{align}&(\!-\lfloor x_j\rfloor_{i,j})_{j\in[d_1]},\quad(\!-\lfloor x_j+\phi_j(x_{\le d_1},-\lfloor x_{\le d_1}\rfloor_i)\rfloor_{i,j})_{d_1+1\le j\le d_2},\notag\\&(\!-\lfloor x_j+\varphi_j(x_{\le d_1},-\lfloor x_{\le d_1}\rfloor_i)+\phi_j(x_{\le d_2},x_{\le d_2}^\ast)\rfloor_{i,j})_{d_2+1\le j\le d_3},\end{align}

where $x_{\le d_2}^\ast$ has coordinates equal to those on the first line of (2·2) (so it implicitly depends on i) and $\lfloor x_{\le d_1}\rfloor_i = (\lfloor x_1\rfloor_{i,1},\ldots, \lfloor x_1\rfloor_{i,d_1})$ . Furthermore let $\{x_{\le d_1}\}_i = x - \lfloor x_{\le d_1}\rfloor_i$ . Note that $\gamma\in \Gamma$ by Lemma 2·5.

Thus $x\gamma$ has coordinates

(2·3) \begin{align}&(\{x_j\}_{i,j})_{j\in[d_1]},\quad(\{x_j+\phi_j(x_{\le d_1},-\lfloor x_{\le d_1}\rfloor_i)\}_{i,j})_{d_1+1\le j\le d_2},\notag\\&(\{x_j+\varphi_j(x_{\le d_1},-\lfloor x_{\le d_1}\rfloor_i)+\phi_j(x_{\le d_2},x_{\le d_2}^\ast)\}_{i,j})_{d_2+1\le j\le d}\end{align}

which is in the specified fundamental domain $\prod_{j=1}^d[\beta_j^{(i)}-1/2,\beta_j^{(i)}+1/2)$ for $F_i$ . Recall also that $\phi_j$ for $j\in[d_2+1,d]$ has certain 0 coefficients.

Let $x^{\ast}_{[d_1+1,d_2]}$ denote the vector with $d_2$ coordinates, the first $d_1$ of which are zero and the last $d_2-d_1$ of which are $(\!-\lfloor x_j+\phi_j(x_{\le d_1},-\lfloor x_{\le d_1}\rfloor_i)\rfloor_{i,j})_{d_1+1\le j\le d_2}$ . Let $y_j = x_j+\phi_j(x_{\le d_1},-\lfloor x_{\le d_1}\rfloor_i)$ , $y^{\ast}_{[d_1+1,d_2]}$ analogously be a vector of these coordinates and $d_1$ many 0s, and let $\{y_{[d_1+1,d_2]}\}_i^\ast = (0,\ldots,0,\{y_{d_1+1}\}_{i,d_1+1},\ldots,\{y_{d_2}\}_{i,d_2})$ .

Note that we have the identity

\[-y\lfloor z\rfloor_2=\lfloor y\rfloor_1 z-yz-\lfloor y\rfloor_1\lfloor z\rfloor_2+\{y\}_1\{z\}_2,\]

where the subscripts 1 and 2 indicate potentially different shift types. Therefore

\begin{align*}\phi_j(x_{\le d_2},x_{\le d_2}^\ast) &= \phi_j(x_{\le d_2},-\lfloor x_{\le d_1}\rfloor_i) + \phi_j(x_{\le d_2},x^{\ast}_{[d_1+1,d_2]})\\&= \phi_j(x_{\le d_2},-\lfloor x_{\le d_1}\rfloor_i) + \phi_j(x_{\le d_1},x^{\ast}_{[d_1+1,d_2]})\\&= \phi_j(x_{\le d_2},-\lfloor x_{\le d_1}\rfloor_i) + \phi_j(\lfloor x_{\le d_1}\rfloor_i,y^{\ast}_{[d_1+1,d_2]})-\phi_j(x_{\le d_1},y^{\ast}_{[d_1+1,d_2]}) \\&\qquad\qquad+ \phi_j(\lfloor x_{\le d_1}\rfloor_i, x^{\ast}_{[d_1+1,d_2]})+ \phi_j(\{x_{\le d_1}\}_i,\{y_{[d_1+1,d_2]}\}_i^\ast).\end{align*}

The equality $\phi_j(x_{\le d_2},x^{\ast}_{[d_1+1,d_2]}) = \phi_j(x_{\le d_1},x^{\ast}_{[d_1+1,d_2]})$ follows as $\phi_j$ has no nonzero coordinates on the box $[d_1+1,d_2]^2$ .

This implies that

\begin{align*}&(\{x_j+\varphi_j(x_{\le d_1},-\lfloor x_{\le d_1}\rfloor_i)+\phi_j(x_{\le d_2},x_{\le d_2}^\ast)\}_{i,j})_{d_2+1\le j\le d}\\&= (\{x_j+\varphi_j(x_{\le d_1},-\lfloor x_{\le d_1}\rfloor_i)+\phi_j(x_{\le d_2},-\lfloor x_{\le d_1}\rfloor_i) + \phi_j(\lfloor x_{\le d_1}\rfloor_i,y^{\ast}_{[d_1+1,d_2]})\\&\qquad-\phi_j(x_{\le d_1},y^{\ast}_{[d_1+1,d_2]})+ \phi_j(\{x_{\le d_1}\}_i,\{y_{[d_1+1,d_2]}\}_i^\ast)\}_{i,j})_{d_2+1\le j\le d};\end{align*}

we are able to drop $\phi_j(\lfloor x_{\le d_1}\rfloor_i, x^{\ast}_{[d_1+1,d_2]})$ as $\phi_j$ has integral coefficients and we are taking fractional parts. Using the above general identity and this expression, we thus have coordinates of the form

(2·4) \begin{align}&(\{x_j\}_{i,j})_{j\in[d_1]},\quad(\{x_j-\phi_j(x_{\le d_1},\lfloor x_{\le d_1}\rfloor_i)\}_{i,j})_{d_1+1\le j\le d_2},\notag\\&(\{x_j+\varphi_j^\ast(x_{\le d_1},\lfloor x_{\le d_1}\rfloor_i)+\phi_j^\ast(x_{\le d_2},\lfloor x_{\le d_1}\rfloor_i)\nonumber\\ &\quad +\sigma_j(x_{\le d_2},x_{\le d_1})+\phi_j(\{x_{\le d_1}\}_i,\{y_{[d_1+1,d_2]}\}^{\ast})\}_{i,j})_{d_2+1\le j\le d_3}.\end{align}

Here $\varphi_j^{\ast}$ are degree at most 3 polynomials and $\phi_j^{\ast},\sigma_j$ are bilinear forms. Additionally, all coefficients are integral with heights bounded by $(O(Md))^{O(1)}$ .

Let us briefly take stock of what has been accomplished in the last manipulation. Note that there are no longer any “iterated” floor expressions and the only terms being “floored” are $x_{\le d_1}$ . The Bohr sets we will ultimately take therefore are determined by these coordinates only.

We now “lift” $F_i$ from a function on $G/\Gamma$ to $\widetilde{F}_i$ on $(\mathbb{R}/\mathbb{Z})^d$ . This can be done via identifying $(\mathbb{R}/\mathbb{Z})^d$ with the fundamental domain $\prod_{j=1}^d[\beta_j^{(i)}-1/2,\beta_j^{(i)}+1/2)$ with respect to $\psi_{\exp}$ . Now, $\widetilde{F}_i$ is seen to be $O(LM/\varepsilon)^{O(d^{O(1)})}$ -Lipschitz on $(\mathbb{R}/\mathbb{Z})^d$ by Lemma 2·6. (We are also using that the support of $\widetilde{F}_i$ is close to the center of the torus, so the torus metric and $\ell^\infty$ on $\mathbb{R}^d$ are the same where it matters.)

Therefore it is sufficient to consider $\widetilde{F}_i$ with coordinates given by

(2·5) \begin{align}&(x_j)_{j\in[d_1]},\quad(x_j-\phi_j(x_{\le d_1},\lfloor x_{\le d_1}\rfloor_i))_{d_1+1\le j\le d_2},\notag\\&(x_j+\varphi_j^\ast(x_{\le d_1},\lfloor x_{\le d_1}\rfloor_i)+\phi_j^\ast(x_{\le d_2},\lfloor x_{\le d_1}\rfloor_i)+\sigma_j(x_{\le d_2},x_{\le d_1})\nonumber \\&+\phi_j(\{x_{\le d_1}\}_i,\{y_{[d_1+1,d_2]}\}_i^\ast))_{d_2+1\le j\le d_3}\end{align}

which live on the torus.

We now “smooth” $\{x_{\le d_1}\}_i$ and $\{y_{[d_1+1,d_2]}\}_i^\ast$ . We replace $\{\cdot\}_{i,j}$ with a 1-periodic function $\{\cdot\}_{i,j,\mathrm{sm}}$ which agrees with $\{x\}_{i,j}$ if $\{x\}_{i,j}\in [\beta_j^{(i)}-1/4,\beta_j^{(i)}+1/4)$ and is O(1)-Lipschitz when viewed as a function on $\mathbb{R}/\mathbb{Z}$ . Given this we write $\{x_{\le d_1}\}_{\mathrm{sm}}$ and $\{y_{[d_1+1,d_2]}\}_{\mathrm{sm}}^\ast$ for the associated vectors where smooth fractional parts have been used.

We claim that it suffices to consider $\widetilde{F}_i$ with coordinates given by

(2·6) \begin{align}&(x_j)_{j\in[d_1]},\quad(x_j-\phi_j(x_{\le d_1},\lfloor x_{\le d_1}\rfloor_i))_{d_1+1\le j\le d_2},\notag\\&(x_j+\varphi_j^\ast(x_{\le d_1},\lfloor x_{\le d_1}\rfloor_i)+\phi_j^\ast(x_{\le d_2},\lfloor x_{\le d_1}\rfloor_i)+\sigma_j(x_{\le d_2},x_{\le d_1})\nonumber \\ &\quad +\phi_j(\{x_{\le d_1}\}_{\mathrm{sm}},\{y_{[d_1+1,d_2]}\}_{\mathrm{sm}}^\ast))_{d_2+1\le j\le d_3}.\end{align}

Note that if $\{x_{\le d_1}\}_{\mathrm{sm}}\neq\{x_{\le d_1}\}_i$ or $\{y_{[d_1+1,d_2]}\}_{\mathrm{sm}}\neq\{y_{[d_1+1,d_2]}\}_i^\ast$ we immediately see that one of the coordinates in the first two lines has been forced outside the support of $\widetilde{F}_i$ and thus the value is already by construction zero in both coordinates. Otherwise the coordinates in (2·5) and (2·6) match and the representation has been unchanged.

Step 3. Fourier expansion of nilsequence. Now, $\widetilde{F}_i\colon(\mathbb{R}/\mathbb{Z})^d\to\mathbb{C}$ is a Lipschitz function. Therefore by Lemma 2·7 with parameter $\tau$ ,

\[\sup_{z\in (\mathbb{R}/\mathbb{Z})^d}\bigg|\widetilde{F}_i(z) - \sum_{|\xi|\le O(M\tau^{-1})^{O(d^{O(1)})}}c_{\xi} e(\xi\cdot z)\bigg|\le \tau\]

with $|c_{\xi}|\le O(M\tau^{-1})^{O(d^{O(1)})}$ .

Using this Fourier representation, we have

\begin{align*}&\sup_{x\in G}\bigg|F_i(x\Gamma) - \sum_{|\xi|\le O(M\tau^{-1})^{O(d^{O(1)})}}c_{\xi} e\bigg(\sum_{j=1}^{d}T_{\xi,j}x_j + \widetilde{\varphi}_{\xi}(x_{\le d_1},\lfloor x_{\le d_1}\rfloor_i)+ \widetilde{\phi}_{\xi}(x_{\le d_2},\lfloor x_{\le d_1}\rfloor_i) \\&\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\widetilde{\sigma}_\xi(x_{\le d_1},x_{\le d_2})+ \widetilde{\phi}'_{\xi}(\{x_{\le d_1}\}_{\mathrm{sm}},\{y_{[d_1+1,d_2]}\}_{\mathrm{sm}}^\ast) \bigg) \bigg|\le \tau\end{align*}

such that $\widetilde{\varphi}_\xi$ is at most a degree 3 polynomial, and $\widetilde{\phi}_\xi,\widetilde{\sigma}_\xi,\widetilde{\phi}'_\xi$ are bilinear forms. Furthermore all coefficients involved are integers bounded by $(M\tau^{-1})^{O(d^{O(1)})}$ .

The final smoothing we perform before we specialise to the polynomial sequence under consideration is to remove the $\{\cdot\}_{\mathrm{sm}}\{\cdot\}_{\mathrm{sm}}$ terms. Note that the function

\[(x,y)\to e(T\{x\}_{i,j,\mathrm{sm}}\{y\}_{i,j,\mathrm{sm}})\]

is $O(T L^{O(1)})$ -Lipschitz. We apply Lemma 2·7 with $\varepsilon = (M \tau^{-1})^{-O(d^{O(1)})}$ sufficiently small and replace each of the possible $d_1\cdot (d_2-d_1)$ possible combinations for the “smoothed parts” simultaneously in each term. We find that

(2·7) \begin{align}&\sup_{x\in G}\bigg|F_i(x\Gamma) \nonumber\\[5pt] &- \sum_{k\in I}c_k e\bigg(\sum_{j=1}^{d}T_{k,j}x_j + \widetilde{\varphi}_k(x_{\le d_1},\lfloor x_{\le d_1}\rfloor_i) + \widetilde{\phi}_k(x_{\le d_2},\lfloor x_{\le d_1}\rfloor_i)+\widetilde{\sigma}_k(x_{\le d_2},x_{\le d_1})\bigg) \bigg|\le 2\tau\end{align}

with $|I|\le (M \tau^{-1})^{O(d^{O(1)})}$ , $|c_k|\le (M\tau^{-1})^{O(d^{O(1)})}$ , $\widetilde{\varphi}_k$ is an at most degree 3 polynomial, $\widetilde{\phi}_k$ and $\widetilde{\sigma}_k$ are bilinear forms, and all coefficients of these forms bounded by $(M \tau^{-1})^{O(d^{O(1)})}$ . Explicitly, we used Lemma 2·7 on $(\mathbb{R}/\mathbb{Z})^2$ with parameter $\varepsilon$ many times and multiplied the representations together.

We now specialise to the case of interest. Note that our primary concern is with the case where $x = g(n)$ and therefore $x_j = p_j(n)$ where $\deg p_j\le i$ for $j\le d_i$ if $i\in\{1,2,3\}$ . Using the representation (2·7) and collecting terms we have

\begin{align*}&\sup_{n\in \mathbb{Z}}\bigg|F_i(g(n)\Gamma) - \sum_{k\in I}c_k e(H_k(n)) \bigg|\le 2\tau,\end{align*}

where $H_k\colon\mathbb{Z}\to\mathbb{R}$ is a sum of terms of the form:

1. (i) $\alpha n^3 + \beta n^2 + \gamma n$ ;

2. (ii) $(\alpha n^2 + \beta n + \gamma) \lfloor p_j(n)\rfloor_{i,j}$ for $1\le j\le d_1$ ;

3. (iii) $(\alpha n + \beta)\lfloor p_j(n)\rfloor_{i,j}\lfloor p_k(n)\rfloor_{i,k}$ for $1\le j\le k\le d_1$ ;

4. (vi) $\gamma\lfloor p_j(n)\rfloor_{i,j}\lfloor p_k(n)\rfloor_{i,k}\lfloor p_\ell(n)\rfloor_{i,\ell}$ for $1\le j\le k\le \ell \le d_1$ ;

note that one can always collect terms so that there are at most $d^{O(1)}$ terms in each expression.

Step 4. N-periodicity and introducing Bohr sets. Let $\rho=1/100$ , which will be the radius of our Bohr sets. Now, note that the expressions $e(H_k(n))$ are not “obviously” N-periodic. We will artificially make them so via a partition of unity argument. There exist smooth $\chi_1,\ldots,\chi_{10^{3}}$ such that $\chi_j\colon(\mathbb{R}/\mathbb{Z})\to \mathbb{R}$ are nonnegative with $\operatorname{supp}(\chi_{j})\subseteq[j/10^{3}, (j+2)/10^{3}) \;\mathrm{mod}\;1$ and $\sum_{j=1}^{10^{3}}\chi_j = 1$ . We have

\begin{align*}&F_i(g(n)\Gamma) = \sum_{j\in [10^{3}]}\chi_j(n/N)F_i(g(n)\Gamma),\\\sup_{\substack{n\in \mathbb{Z}\\ j\in [10^{3}]}}\bigg|&F_i(g(n)\Gamma)\chi_j(n/N) - \chi_j(n/N)\sum_{k\in I}c_k e(H_k(n)) \bigg|\le O(\tau).\end{align*}

Now fix $j\in [10^{3}]$ ; such a term only contributes when $n/N \in [j/10^{3}, (j+2)/10^{3}) \mod 1$ . Next note that each $p_\ell(n) = \alpha_\ell n$ for $\ell\le d_1$ and because the initial nilsequence $g(n)\Gamma$ is N-periodic, by [ 23 , lemma A·12] we have that $\alpha_\ell \in (1/N)\mathbb{Z}$ .

Note that in order for $\chi_j(n/N)F_i(g(n)\Gamma)$ to be nonzero, we need that:

1. (i) $n/N \equiv [j/10^{3}, (j+2)/10^{3}) \mod 1$ ;

2. (ii) $\{p_\ell(n)\}_{i,\ell} \in [\beta_\ell^{(i)}-10^{-3},\beta_\ell^{(i)}+10^{-3})$ for all $\ell\in[d_1]$ .

These conditions are clearly N-periodic and in fact all such n lie in a shifted Bohr set with frequency set $S = \{1/N\} \cup \{\alpha_\ell\}_{1\le \ell\le d_1}\subseteq(1/N)\mathbb{Z}$ . If the corresponding shifted Bohr set is empty we know that $\chi_j(n/N)F_i(g(n)\Gamma) = 0$ and we approximate $\chi_j(n/N)F_i(g(n)\Gamma)$ by 0. Else there is $y_j\in\mathbb{Z}/N\mathbb{Z}$ such that $y_j/N \equiv [j/10^{3}, (j+2)/10^{3}) \mod 1$ and $\{\alpha_\ell y_j\}_{i,\ell} \in [\beta_\ell^{(i)}-10^{-3},\beta_\ell^{(i)}+10^{-3})$ for $\ell\in[d_1]$ . Letting $J_i$ be the set of j for which this shifted Bohr set is nonempty, we have

\[\sup_{\substack{n\in \mathbb{Z}\\ j\in J_i}}\bigg|F_i(g(n)\Gamma)\chi_j(n/N) - \mathbb{1}_{n\in y_j + B(S, \rho)}\chi_j(n/N)\sum_{k\in I}c_k e(H_k(n)) \bigg|\le O(\tau).\]

We now apply Fourier expansion on $\chi_j$ using Lemma 2·7 on the torus $\mathbb{R}/\mathbb{Z}$ and appropriately chosen error parameter $\tau' = (M\tau^{-1})^{-O(d^{O(1)})}$ . We find

(2·8) \begin{equation}\sup_{\substack{n\in\mathbb{Z}\\j\in J_i}}\bigg|F_i(g(n)\Gamma)\chi_j(n/N) - \mathbb{1}_{n\in y_j + B(S,\rho)}\chi_j(n/N)\sum_{k\in I, |\xi|\le (\tau')^{-O(1)}}c_kd_{\xi} e(H_k(n) + \xi n/N) \bigg|\le O(\tau)\end{equation}

with $|d_{\xi}|\le (\tau')^{-O(1)}$ .

Given $n\in y_j + B(S,\rho)$ , we will now replace $H_k(n)$ by $H_{k,j}(n)$ so that the latter is N-periodic (and it will be locally cubic on $y_j+B(S,\rho)$ ). We fix an interval $I_j \in [\!-\!N,N)$ of integers of length at most $\lceil N/50\rceil$ such that for each $n\in y_j + B(S,\rho)$ there is a unique integer $n'\in I_j$ such that $n'\equiv n \;\mathrm{mod}\;N$ . This exists since $1/N\in S$ . We then define $H_{k,j}(n) \equiv H_k(n') \mod 1$ for such $n\in y_j + B(S,\rho)$ and $H_{k,j}(n) = 0$ otherwise; note that $H_{k,j}(n)$ is taking values in $\mathbb{R}/\mathbb{Z}$ which is sufficient as we are plugging these values into the exponential function.

It is easy to see that (2·8) holds with $H_k(n)+\xi\cdot n/N$ replaced by $H_{k,j}(n)$ since everything in the inequality except for potentially $H_k(n)$ is N-periodic (recall $S\subseteq(1/N)\mathbb{Z}$ ), and there is a supremum over $n\in\mathbb{Z}$ on the outside. So, we have

\[\sup_{\substack{n\in \mathbb{Z}\\ j\in J_i}}\bigg|F_i(g(n)\Gamma)\chi_j(n/N) - \mathbb{1}_{n\in y_j + B(S, \rho)}\sum_{k\in I, |\xi|\le (\tau')^{-O(1)}}c_k d_{\xi} e(H_{k,j}(n)+\xi n/N) \bigg|\le O(\tau),\]

where $H_{k,j}(n)$ is N-periodic by construction. Taking $\tau = \eta M^{-O(d^{O(1)})}$ and summing over j and i then finishes the proof modulo showing that $H_{k,j}(n)$ is a locally cubic function on $y_j + B(S,\rho)$ .

Step 5. Local cubicity on shifted Bohr sets. Fix i, $j\in J_i$ , and $k\in I$ . We wish to show local cubicity of $H_{k,j}$ on $y_j+B(S,\rho)$ . Suppose that $n,h_1,h_2,h_3\in\mathbb{Z}/N\mathbb{Z}$ are such that $n + \sum_{\ell=1}^{3}\epsilon_\ell h_\ell\in y_j + B(S,\rho)$ for all $(\epsilon_1,\epsilon_2,\epsilon_3)\in \{0,1\}^3$ . We consider the representatives of $n + \sum_{\ell=1}^{3}\epsilon_\ell h_\ell$ in $I_j$ ; we see that there exist $n',h_1',h_2',h_3'\in \mathbb{Z}$ such that $n' + \sum_{\ell=1}^{3}\epsilon_\ell h_\ell' \equiv n + \sum_{\ell=1}^{3}\epsilon_\ell h_\ell \;\mathrm{mod}\;N$ and $n' + \sum_{\ell=1}^{3}\epsilon_\ell h_\ell'\in I_j$ for all $\epsilon\in\{0,1\}^3$ . Indeed, one can look at the representatives of each $n + \sum_{\ell=1}^{3}\epsilon_\ell h_\ell$ in $I_j$ , and use the fact that if $t_1 + t_2 \equiv t_3 + t_4 \;\mathrm{mod}\;N$ with $t_i\in I_j$ then $t_1 + t_2 = t_3 + t_4$ (since $I_j$ has length at most $\lceil N/50\rceil$ ).

Therefore it suffices to prove that $H_k\colon\mathbb{Z}\to\mathbb{R}$ is locally cubic on $y_j+B(S,\rho)$ . Recalling the classification of $H_k(n)$ as a sum of terms of various kinds, it suffices to verify this for each individual function type which is combined to give $H_k(n)$ .

Note that pure polynomials are always the correct degree (on all of $\mathbb{Z}$ ). By Lemma 2·8 it suffices to verify that $\lfloor \alpha_\ell n \rfloor_{i,\ell}$ is locally linear on $y_j+B(S,\rho)$ for $1\le\ell\le d_1$ .

Given $n,n + h_1, n+h_2, n+h_1+h_2\in y_j + B(S,\rho)$ we have that

\begin{align*}&\lfloor \alpha_\ell n \rfloor_{i,\ell} - \lfloor \alpha_\ell (n+h_1) \rfloor_{i,\ell} - \lfloor \alpha_\ell (n+h_2) \rfloor_{i,\ell} + \lfloor \alpha_\ell (n+h_1+h_2) \rfloor_{i,\ell} \in \mathbb{Z},\\&\big|\lfloor \alpha_\ell n \rfloor_{i,\ell} - \lfloor \alpha_\ell (n+h_1) \rfloor_{i,\ell} - \lfloor \alpha_\ell (n+h_2) \rfloor_{i,\ell} + \lfloor \alpha_\ell (n+h_1+h_2) \rfloor_{i,\ell} \big|\\&= \big|\{ \alpha_\ell n \}_{i,\ell} - \{ \alpha_\ell (n+h_1) \}_{i,\ell} - \{ \alpha_\ell (n+h_2) \}_{i,\ell} + \{ \alpha_\ell (n+h_1+h_2) \}_{i,\ell} \big|\le 1/2,\end{align*}

the inequality using that $\rho=1/100$ and every $\alpha_\ell(n+\epsilon_1h_1+\epsilon_2h_2)$ must be close in $\mathbb{R}/\mathbb{Z}$ to $\alpha_\ell y_j$ . Additionally, we recall $\{\alpha_\ell y_j\}\in[\beta_\ell^{(i)}-10^{-3},\beta_\ell^{(i)}+10^{-3})$ which implies that there is no discontinuity in $\{\cdot\}_{i,\ell}$ in the area of consideration.

Therefore

\[\lfloor \alpha_\ell n \rfloor_{i,\ell} - \lfloor \alpha_\ell (n+h_1) \rfloor_{i,\ell} - \lfloor \alpha_\ell (n+h_2) \rfloor_{i,\ell} + \lfloor \alpha_\ell (n+h_1+h_2) \rfloor_{i,\ell} = 0\]

for $n,n + h_1, n+h_2, n+h_1+h_2\in y_j + B(S,\rho)$ as desired. This (finally) completes the proof.

3. Proof of Theorem 1·1 given Proposition 1·4

In this section, we convert Proposition 1·4 into a density increment using the strategy of Heath-Brown [ 16 ] and Szemerédi [ 35 ]. Our treatment is closely modeled on that of Green and Tao [ 13 ]; the crucial idea is that one may group together a large number of phases before passing to a subprogression.

Throughout this section we will consider functions $f\colon[N']\to[\!-\!1,1]$ corresponding to sets and shifted indicators. By Bertrand’s postulate, we may find a prime N such that $1024 N' \le N \le 2048 N'$ . We may thus embed [N ^′] inside $\mathbb{Z}/(N\mathbb{Z})$ and lift f to $\mathbb{Z}/N\mathbb{Z}$ (mapping inputs not congruent to an element of [N ^′] to 0). We define the quintilinear operator

\[\Lambda(f_1,\ldots,f_5) = \mathbb{E}_{x,y\in\mathbb{Z}/N\mathbb{Z}} \prod_{k=1}^{5}f_k(x+(k-1)y)\text{ and }\Lambda(f) = \Lambda(f,f,f,f,f).\]

We now state the key claim for this section, which we will prove in Section 3.3

Proposition 3·1. Fix a constant $c \gt 0$ . There exist $c' \gt 0$ and $C \gt 0$ such that the following holds. Consider a function $f\colon[N']\to [0,1]$ such that $\mathbb{E}_{n\in [N']}f(n) = \delta \gt 0$ and a prime N such that $1024 N'\le N\le 2048 N'$ . Let $M(\delta) = \exp\!((\!\log\!(1/\delta))^C)$ . At least one of the following possibilities always holds:

1. (i) $N'\le\exp\!(M(\delta))$ ;

2. (ii) $\big|\Lambda(f) - \Lambda(\delta \mathbb{1}_{[N']})\big| \le c \delta^{5}$ ;

3. (iii) there exists an arithmetic progression P of length at least $N^{1/M(\delta)}$ such that

   \[\mathbb{E}_{n\in P}f(n)\ge (1+c')\delta.\]

With this, we can prove the main result.

Proof of Theorem 1·1 given Proposition 3·1. Suppose that $A\subseteq [N]$ has size $\delta N$ and has no 5-term arithmetic progressions. We now perform the density increment strategy using $A_0 = A$ , $N_0' = N$ , and $\delta_0 = \delta$ .

For each $N_i'$ choose a prime $N_i$ between $1024 N_i'\le N_i\le 2048 N_i'$ . If $N_i'\le M(\delta_i)$ , we immediately terminate. Else note that

\[\big|\Lambda(\mathbb{1}_{A_i}) - \Lambda(\delta_i\mathbb{1}_{[N_i']})\big| \ge |\Lambda(\delta_i\mathbb{1}_{[N_i']})\big| - |A_i|N_i^{-2}\gg \delta_i^{5}\]

as $A_i$ is free of 5-term arithmetic progressions. Therefore the third case in the trichotomy must hold and there exists a long progression $P_i$ on which the density of $A_i$ increases by a factor of at least $(1+c')$ . Let $A_{i+1}$ be $A_i\cap P_i$ rescaled so that $P_i$ starts at 0 and has common difference 1, let $N_{i+1}' = |P_i|$ , and let $\delta_{i+1} = |A_i\cap P_i|/|P_i|\ge(1+c')\delta_i$ .

Since the density of a set cannot exceed 1, we must terminate in at most $O(\!\log\!(1/\delta))$ iterations. If we terminate at i, we must have

\[N^{{(1/M(\delta))}^{O(\!\log\!(1/\delta))}}\le N_i'\le\exp\!(M(\delta_i))\le\exp\!(M(\delta)).\]

Rearranging this gives exactly that

\[\log N \le {M(\delta)}^{O(\!\log\!(1/\delta))}\]

or, as desired,

\[\delta \ll \frac{1}{\exp\!((\!\log\log N)^{\Omega(1)})}.\]

3·1. Inverse theorem relative to linear and cubic factors

We introduce a framework for studying functions satisfying a correlation as given by Proposition 1·4, considering a $\sigma$ -algebra (or factor) which incorporates the information of the approximate value of our linear and cubic functions on [N]. Our treatment closely follows that of Green and Tao [ 13 ] (and uses elements from Peluse and Prendiville [ 25 ]).

Definition 3·2. We define a factor $\mathcal{B}$ of $\mathbb{Z}/N\mathbb{Z}$ to be a partition $\mathbb{Z}/N\mathbb{Z} = \bigsqcup_{B\in \mathcal{B}}B$ . We define $\mathcal{B}(x)$ for $x\in\mathbb{Z}/N\mathbb{Z}$ to be the part of $\mathcal{B}$ that contains x.

We say $\mathcal{B}'$ refines $\mathcal{B}$ if every part of $\mathcal{B}$ can be written as a disjoint union of parts of $\mathcal{B}'$ . We define a join of a sequence of factors to be the partition (discarding empty parts)

\[\mathcal{B}_1\vee\cdots\vee\mathcal{B}_d \;:\!=\; \{B_1\cap\cdots\cap B_d\colon B_i\in \mathcal{B}_i\}.\]

Define a function $\phi\colon S\to\mathbb{R}/\mathbb{Z}$ for $S\subseteq\mathbb{Z}/N\mathbb{Z}$ to be irrational if $\phi$ takes on irrational values. Define the factor $\mathcal{B}_{\phi,K}$ with respect to $\phi$ of resolution K as the partition

\[((\mathbb{Z}/N\mathbb{Z})\setminus S)\sqcup \bigsqcup_{0\le j\le K-1}\{x\in\mathbb{Z}/N\mathbb{Z}\colon\lVert\phi(x) - j/K\rVert_{\mathbb{R}/\mathbb{Z}}\le 1/(2K)\}.\]

(Since $\phi$ is irrational this is indeed a disjoint partition.) We further say that $\phi$ respects a factor $\mathcal{B}$ if $\mathcal{B}$ refines $S\sqcup ((\mathbb{Z}/N\mathbb{Z})\setminus S)$ .

We define a factor of complexity $(d_1,d_2)$ and resolution K via the data of irrational linear functions $\phi_1,\ldots,\phi_{d_1}$ (defined on $\mathbb{Z}/N\mathbb{Z}$ ) and irrational locally cubic functions $\phi_1',\ldots,\phi_{d_2}'$ (defined on subsets of $\mathbb{Z}/N\mathbb{Z}$ ) which respect $\bigvee_{1\le j\le d_1}\mathcal{B}_{\phi_j,K}$ . The associated factor is $\bigvee_{1\le j\le d_1}\mathcal{B}_{\phi_j,K} \vee \bigvee_{1\le j\le d_1}\mathcal{B}_{\phi_j',K}$ .

Finally, given a factor $\mathcal{B}$ we define $\Pi_{\mathcal{B}}f$ by

\[\Pi_{\mathcal{B}}f(x) = \mathbb{E}_{y\in\mathcal{B}(x)}f(y).\]

Remark. We ensure all functions $\phi$ we consider are irrational in order to avoid issues regarding the hitting exactly the boundary of the factor. Note that if a locally cubic function $\phi$ respects a partition $\mathcal{B}$ then we see that the support set is in the $\sigma$ -algebra generated by $\mathcal{B}$ . In particular, we can treat $\phi$ as either “undefined” on an atom $\mathcal{B}$ or as locally cubic on the associated atom.

We now restate Proposition 1·4 in the language of factors.

Lemma 3·3. There exists $C \gt 0$ such that the following holds. Fix $\eta \gt 0$ and let f be a function $f\colon\mathbb{Z}/N\mathbb{Z}\to[0,1]$ such that $\lVert f\rVert_{U^4(\mathbb{Z}/N\mathbb{Z})}\ge\eta$ .

Let $M(\eta) = \exp\!((\!\log\!(1/\eta))^C)$ and fix an integer $K\ge \exp\!(M(\eta))$ . Suppose that $N\ge 8K$ . There exist $d\le M(\eta)$ and a factor $\mathcal{B}$ of complexity (d,1) and resolution K such that

\[\lVert\Pi_{\mathcal{B}}f\rVert_{L^{1}(\mathbb{Z}/N\mathbb{Z})} \ge \exp\!(\!-\!M(\eta)).\]

Proof. By Proposition 1·4, there exist $\rho=1/100$ , $S\subseteq (1/N)\mathbb{Z}$ with $|S|\ll(\!\log\!(1/\eta))^{C}$ (say $|S|=d$ ), and $y\in \mathbb{Z}/N\mathbb{Z}$ such that $B=B(S,\rho)$ is a Bohr set and $\phi\colon y+B\to\mathbb{R}/\mathbb{Z}$ is a locally cubic phase function such that

\[|\mathbb{E}_{t\in\mathbb{Z}/N\mathbb{Z}}\mathbb{1}_{t\in y + B}f(t)e(\!-\!\phi(t))|\ge \exp\!(\!-\!(\!\log\!(1/\eta))^C/2).\]

Let C ^′ be a sufficiently large constant and assume $K \ge \exp\!((\!\log\!(1/\eta))^{C'})$ . Take $\phi_i\colon\mathbb{Z}/N\mathbb{Z}\to \mathbb{R}/\mathbb{Z}$ to be $\phi_i(x) = a_ix - a_iy + \alpha$ where $\alpha$ is an irrational which is smaller than $(2K)^{-1}$ and $a_i\in S$ .

Note that $y + B$ is exactly the set

\[T_1 = \Big\{x\in\mathbb{Z}/N\mathbb{Z}\colon \sup_{a_i\in S}\lVert a_ix - a_i y\rVert_{\mathbb{R}/\mathbb{Z}}\le\rho\Big\}.\]

Note that the set

\[T_2 = \Big\{x\in\mathbb{Z}/N\mathbb{Z}\colon\sup_{a_i\in S}\lVert a_ix - a_iy + \alpha\rVert_{\mathbb{R}/\mathbb{Z}}\le (2K)^{-1}\cdot (2\lfloor K \rho\rfloor-1)\Big\}\]

is measurable with respect to the factor $\bigvee_{1\le j\le d}\mathcal{B}_{\phi_j,K}$ . Also, $T_2\subseteq T_1$ since $|\alpha|\le 1/(2K)$ and $(2K)^{-1}\cdot (2\lfloor K \rho\rfloor-1) + (2K)^{-1}\le\rho$ . Furthermore we have

\[T_1\setminus T_2 \subseteq \Big\{x\in\mathbb{Z}/N\mathbb{Z}\colon\sup_{a_i\in S}\lVert a_ix - a_i y\rVert_{\mathbb{R}/\mathbb{Z}}\in[\rho-2/K,\rho)\Big\}\]

and thus we have

\[|T_2\setminus T_1|\le |S|\cdot(5/K)N= 5dN/K\]

as long as $8K\le N$ . Here we have used that $S\subseteq\mathbb{Z}/N\mathbb{Z}$ where N is prime.

So, since C ^′ is large with respect to C and f is 1-bounded we have

\[|\mathbb{E}_{t\in\mathbb{Z}/N\mathbb{Z}}\mathbb{1}_{t\in T_2}f(t)e(\!-\!\phi(t))|\gg \exp\!(\!-\!(\!\log\!(1/\eta))^C/2).\]

The locally cubic function we will consider is $\phi^{\ast}\colon T_2\to \mathbb{R}/\mathbb{Z}$ given by $\phi^{\ast}(x) = \phi(x) + \alpha$ . This is well-defined as $T_2\subseteq T_1 = y+B$ . Let $\mathcal{B} = \bigvee_{1\le j\le d}\mathcal{B}_{\phi_j,K} \vee \mathcal{B}_{\phi^\ast,K}$ and note that

\[\sup_{x\in \mathbb{Z}/N\mathbb{Z}}\big|e(\!-\!\phi(x)) - \Pi_{\mathcal{B}}[e(\!-\!\phi^{\ast}(x))]\big|\le 2\pi/K\]

since $z\mapsto e(z)$ is a $2\pi$ -Lipschitz function on $\mathbb{R}/\mathbb{Z}$ . Therefore

\[|\mathbb{E}_{t\in\mathbb{Z}/N\mathbb{Z}}f(t)\Pi_{\mathcal{B}}[e(\!-\!\phi^{\ast}(x))]|\ge\exp\!(\!-\!(\!\log\!(1/\eta))^C).\]

$\Pi_{\mathcal{B}}$ is self-adjoint with respect to the standard inner product (see e.g. [ 25 , lemma 4·3(ii)]) so

\begin{align*}\mathbb{E}_{t\in\mathbb{Z}/N\mathbb{Z}}|\Pi_{\mathcal{B}}[f(t)]|&\ge|\mathbb{E}_{t\in\mathbb{Z}/N\mathbb{Z}}\Pi_{\mathcal{B}}[f(t)]e(\!-\!\phi^{\ast}(t))|=|\mathbb{E}_{t\in\mathbb{Z}/N\mathbb{Z}}f(t)\Pi_{\mathcal{B}}[e(\!-\!\phi^{\ast}(t))]|\\&\ge\exp\!(\!-\!(\!\log\!(1/\eta))^C).\end{align*}

This gives the desired result.

We now prove an associated Koopman–von Neumann theorem; our proof is closely modeled after [ 13 , theorem 5·6].

Lemma 3·4. There exists $C \gt 0$ such that the following holds. Fix $\eta \gt 0$ , let f be a function $f\colon\mathbb{Z}/N\mathbb{Z}\to [0,1]$ , and let $\mathcal{B}^{\ast}$ denote an initial factor.

Let $M(\eta) = \exp\!(\!\log\!(1/\eta)^C)$ and let $N\ge 10 M(\eta)$ . There exist $d_1,d_2\le M(\eta)$ such that the following holds. There exists an integer $K\le M(\eta)$ and a factor of $\mathcal{B}'$ of complexity $(d_1,d_2)$ and resolution K such that if $\mathcal{B} = \mathcal{B}^{\ast} \vee \mathcal{B}'$ then

\[\lVert f - \Pi_{\mathcal{B}}f\rVert_{U^4(\mathbb{Z}/N\mathbb{Z})} \le \eta.\]

Proof. We create the factor $\mathcal{B}'$ as a join given by applying Lemma 3·3 iteratively. Set $\mathcal{B}_0 = \mathcal{B}^{\ast}$ .

1. (i) If $\lVert f-\Pi_{\mathcal{B}_i} f\rVert_{U^4(\mathbb{Z}/N\mathbb{Z})}\le \eta$ , we terminate. Set $i^{\ast} = i$ and output $\mathcal{B}'=\bigvee_{0\le i' \lt i^\ast}\mathcal{B}_{i'}'$ .

2. (ii) If $\lVert f-\Pi_{\mathcal{B}_i} f\rVert_{U^4(\mathbb{Z}/N\mathbb{Z})}\ge \eta$ , then set $K = \lceil \exp\!(M(\eta))\rceil$ . By Lemma 3·3 there exists $\mathcal{B}_i'$ such that

   \[\lVert\Pi_{\mathcal{B}_i'}(f-\Pi_{\mathcal{B}_i} f)\rVert_{L^1(\mathbb{Z}/N\mathbb{Z})}\ge \exp\!(\!-\!(\!\log\!(1/\eta))^{C'}).\]
   Here $\mathcal{B}_i' = \bigvee_{1\le j\le d_i}\mathcal{B}_{\phi_j^{(i)},K} \vee \mathcal{B}_{\phi_i',K}$ where $\phi_j^{(i)}$ (for $j\le d_i\le\exp\!((\!\log\!(1/\eta))^{O(1)}))$ ) are irrational linear functions on $\mathbb{Z}/N\mathbb{Z}$ and $\phi_i'$ is a locally cubic function respecting the factor $\bigvee_{1\le j\le d}\mathcal{B}_{\phi_j,K}$ . We set $\mathcal{B}_{i+1} = \mathcal{B}_i\vee \mathcal{B}_i'$ .

Note that $\mathcal{B}=\mathcal{B}^\ast\vee\mathcal{B}'=\mathcal{B}_{i^\ast}$ . Observe that

\begin{align*}\lVert\Pi_{\mathcal{B}_i'}(f-\Pi_{\mathcal{B}_i}f)\rVert_{L^1(\mathbb{Z}/N\mathbb{Z})}&\le\lVert\Pi_{\mathcal{B}_i'}(f-\Pi_{\mathcal{B}_i}f)\rVert_{L^2(\mathbb{Z}/N\mathbb{Z})}=\lVert\Pi_{\mathcal{B}_i'}\Pi_{\mathcal{B}_{i+1}}(f-\Pi_{\mathcal{B}_i}f)\rVert_{L^2(\mathbb{Z}/N\mathbb{Z})}\\&\le\lVert\Pi_{\mathcal{B}_{i+1}}(f-\Pi_{\mathcal{B}_i}f)\rVert_{L^2(\mathbb{Z}/N\mathbb{Z})}=\lVert\Pi_{\mathcal{B}_{i+1}}f-\Pi_{\mathcal{B}_i}f\rVert_{L^2(\mathbb{Z}/N\mathbb{Z})}\\&=(\lVert\Pi_{\mathcal{B}_{i+1}}f\rVert_{L^2(\mathbb{Z}/N\mathbb{Z})}^2-\lVert\Pi_{\mathcal{B}_i}f\rVert_{L^2(\mathbb{Z}/N\mathbb{Z})}^2)^{1/2}.\end{align*}

The final equality is the Pythagorean theorem with respect to projections (this follows from e.g. [ 25 , lemma 4·3(iv)]).

Thus $\lVert\Pi_{\mathcal{B}_{i+1}f}\rVert_{L^2(\mathbb{Z}/N\mathbb{Z})}^2-\lVert\Pi_{\mathcal{B}_i}f\rVert_{L^2(\mathbb{Z}/N\mathbb{Z})}^2\ge \exp\!(\!-\!(\!\log\!(1/\eta))^{O(1)})$ and hence the iteration lasts only $\exp\!((\!\log\!(1/\eta))^{O(1)})$ steps. Therefore the final $\mathcal{B}'$ is generated by at most $\exp\!((\!\log\!(1/\eta))^{O(1)})$ linear functions and a similar number of locally cubic functions. This completes the proof.

3·2. Density increment onto cubic Bohr set

We next need that $\Lambda$ is controlled by the $U^4$ -norm and the $L^{1}$ -norm. The proof is by now standard and hence is omitted (see [ 13 , lemma 3·2] and [ 9 , theorem 3·2]).

Lemma 3·5. Let N be prime and $f_i\colon\mathbb{Z}/N\mathbb{Z}\to \mathbb{C}$ . Then we have:

\begin{align*}|\Lambda(f_1,f_2,f_3,f_4,f_5)|&\le \min_{1\le i\le 5}\lVert f_i\rVert_{L^1(\mathbb{Z}/N\mathbb{Z})}\prod_{j\neq i}\lVert f_j\rVert_{L^\infty(\mathbb{Z}/N\mathbb{Z})};\\|\Lambda(f_1,f_2,f_3,f_4,f_5)|&\le \min_{1\le i \le 5}\lVert f_i\rVert_{U^4(\mathbb{Z}/N\mathbb{Z})}\prod_{j\neq i}\lVert f_j\rVert_{L^\infty(\mathbb{Z}/N\mathbb{Z})}.\end{align*}

Given this we may now prove that there exists a density increment onto a piece of the partition given by Lemma 3·4. The proof is identical to [ 13 , lemma 5·8].

Proposition 3·6. Fix $c \gt 0$ . There exist $C \gt 0$ and $c' \gt 0$ such that the following holds. Given a function $f\colon[N']\to[0,1]$ such that $\mathbb{E}_{n\in[N']}f(n) = \delta \gt 0$ , extend it by 0s to a function on $\mathbb{Z}/N\mathbb{Z}$ where N is prime with $1024N'\le N\le 2048N'$ . Suppose that

\[|\Lambda(f) - \Lambda(\delta \mathbb{1}_{[N']})|\ge c\delta^{5}.\]

Let $\mathcal{B}^{\ast} = [N']\sqcup ((\mathbb{Z}/N\mathbb{Z})\setminus [N'])$ be an initial factor. Let $M(\delta) = \exp\!((\!\log\!(1/\delta))^C)$ and suppose that $N\ge M(\delta)$ . Then there exist $d_1,d_2,K\le M(\delta)$ , a factor $\mathcal{B}$ of complexity $(d_1,d_2)$ and resolution K, and an atom $\Omega^\ast$ of $\mathcal{B}\vee\mathcal{B}^{\ast}$ such that:

1. (i) $\mathbb{P}_{n\in \mathbb{Z}/N\mathbb{Z}}[n\in\Omega^\ast]\ge \exp\!(\!-\!\exp\!((\!\log\!(1/\delta))^C))$ ;

2. (ii) $\mathbb{E}[f(n)|n\in\Omega^\ast]\ge (1+c')\delta$ .

Proof. We may clearly assume $\delta$ is sufficiently small. By Lemma 3·4, there exists a factor $\mathcal{B}$ of complexity $(d_1,d_2)$ and resolution K with $d_1,d_2,K\le \exp\!((\!\log\!(1/\delta))^{O(1)})$ such that

\[\lVert f - \Pi_{\mathcal{B}\vee \mathcal{B}^{\ast}}f\rVert_{U^4(\mathbb{Z}/N\mathbb{Z})}\le c\delta^{5}/10.\]

By telescoping and the second part of Lemma 3·5 we have

\[\big|\Lambda(f) - \Lambda(\Pi_{\mathcal{B}\vee \mathcal{B}^{\ast}}f)\big|\le 5\lVert f-\Pi_{\mathcal{B}\vee \mathcal{B}^{\ast}}f\rVert_{U^4(\mathbb{Z}/N\mathbb{Z})}\le c\delta^{5}/2.\]

Therefore we have

\[|\Lambda(\Pi_{\mathcal{B}\vee \mathcal{B}^{\ast}}f) - \Lambda(\delta \mathbb{1}_{[N']})|\ge c\delta^{5}/2.\]

Let $\mathcal{B}' = \mathcal{B}\vee \mathcal{B}^{\ast}$ and $g = \min(\Pi_{\mathcal{B}'}f,(1+c')\delta)$ and assume that $c'\le \max(c,1)/10^{5}$ . Let $\Omega' = \{x\in\mathbb{Z}/N\mathbb{Z}\colon g(x)\neq \Pi_{\mathcal{B}'}f(x)\}$ and notice that

\begin{align*}|\Lambda(\Pi_{\mathcal{B}'}f) - \Lambda(g)|&\le 5\lVert\Pi_{\mathcal{B}'}f - g\rVert_{L^1(\mathbb{Z}/N\mathbb{Z})} \le 5 \mathbb{P}[n\in \Omega']\\|\Lambda(\delta\mathbb{1}_{[N']}) - \Lambda(g)|&\le 5(1+c')^4\delta^{4}\lVert\delta\mathbb{1}_{[N']}-g\rVert_{L^1(\mathbb{Z}/N\mathbb{Z})}\\\lVert g - \delta \mathbb{1}_{[N']}\rVert_{L^1(\mathbb{Z}/N\mathbb{Z})}&\le \mathbb{P}[n\in \Omega'] + \lVert\delta \mathbb{1}_{[N']} - \Pi_{\mathcal{B}'}f\rVert_{L^1(\mathbb{Z}/N\mathbb{Z})}.\end{align*}

The first and second inequality follow from the first part of Lemma 3·5 while the final inequality follows from the triangle inequality.

Suppose that the proposition is false. Note that there are at most $2(2K)^{d_1+d_2}$ atoms of $\mathcal{B}'$ and define an atom to be small if it has measure at most $(2K)^{-d_1-d_2}\cdot (c\delta^{5})/40$ . We therefore have that on every atom which is not small,

\[\Pi_{\mathcal{B}'}f\le (1+c')\delta\]

(else we take $\Omega^\ast$ to be that atom) and therefore $\mathbb{P}[n\in \Omega']\le c\delta^{5}/20$ . Thus by the first inequality above and the triangle inequality we find

\[|\Lambda(\delta\mathbb{1}_{[N']}) - \Lambda(g)|\ge|\Lambda(\delta\mathbb{1}_{[N']}) - \Lambda(\Pi_{\mathcal{B}'}f)|-5\mathbb{P}[n\in\Omega^\ast]\ge c\delta^{5}/4.\]

By the second inequality, this implies that

\[\lVert\delta\mathbb{1}_{[N']}-g\rVert_{L^1(\mathbb{Z}/N\mathbb{Z})}\ge c\delta/100.\]

By the third inequality we have that

\[\lVert\delta \mathbb{1}_{[N']} - \Pi_{\mathcal{B'}}f\rVert_{L^1(\mathbb{Z}/N\mathbb{Z})}\ge c\delta/200.\]

For a function h let $h_{+} = \max(h,0)$ . For mean zero functions, $\lVert h\rVert_{L^1(\mathbb{Z}/N\mathbb{Z})} = 2\lVert h_{+}\rVert_{L^1(\mathbb{Z}/N\mathbb{Z})}$ . This allows us to derive the contradiction since

\begin{align*}\lVert\delta \mathbb{1}_{[N']} - \Pi_{\mathcal{B'}}f\rVert_{L^1(\mathbb{Z}/N\mathbb{Z})} &= 2\lVert(\Pi_{\mathcal{B'}}f-\delta \mathbb{1}_{[N']})_{+}\rVert_{L^1(\mathbb{Z}/N\mathbb{Z})}\\&\le 2c'\delta + 2((2K)^{-d_1-d_2}\cdot (c\delta^{5})/40)(2(2K)^{d_1+d_2})\le 4c'\delta \lt c\delta/200.\end{align*}

3·3. Density increment onto subprogression

We now finish the argument by partitioning factors in our cubic Bohr partition into long progressions. This is a technical modification of an argument of Green and Tao [ 13 ] for multiple quadratics (the case of a single polynomial is handled in Gowers [ 9 ]); as input they must provide, in the case of quadratics, an explicit implicit constant for a result of Schmidt [ 32 ] which provides recurrence for multiple polynomials mod 1 simultaneously. We require the analogous result for arbitrary degrees.

Proposition 3·7. Fix an integer $k\ge 1$ . There exist $c = c(k) \gt 0$ such that the following holds. Let $\alpha_1,\ldots,\alpha_d$ be real numbers. Then

\[\min_{1\le n\le N}\max_{1\le i\le d}\lVert\alpha_i n^k\rVert_{\mathbb{R}/\mathbb{Z}}\ll_k dN^{-c/d^2}.\]

We defer the proof of this input to Appendix B; it is an essentially verbatim copy of [ 13 , appendix A].

Proof of Proposition 3·1. Step 1. Setup. Let $M=\exp\!((\!\log\!(1/\delta))^C)$ where $C \gt 0$ will be chosen later. Let us assume $N' \gt \exp\!(M(\delta))$ and $|\Lambda(f)-\Lambda(\delta\mathbb{1}_{[N']})| \gt c\delta^5$ . Our aim is to find a progression of length at least $N^{1/M(\delta)}$ where our density is incremented.

Let $\mathcal{B}^{\ast} = [N']\sqcup ((\mathbb{Z}/N\mathbb{Z})\setminus [N'])$ be an initial factor. Let $M'(\delta) = \exp\!((\!\log\!(1/\delta))^{C'})$ where C ^′ is chosen appropriately so as to apply Proposition 3·6. By assumption we have $N\ge M'(\delta)$ . So there exist $d_1,d_2,K\le M'(\delta)$ , a factor $\mathcal{B}$ of complexity $(d_1,d_2)$ and resolution K, and an atom $\Omega^\ast$ of $\mathcal{B}\vee\mathcal{B}^{\ast}$ such that:

1. (i) $\mathbb{P}_{n\in \mathbb{Z}/N\mathbb{Z}}[n\in\Omega^\ast]\ge\exp\!(\!-\!\exp\!((\!\log\!(1/\delta))^{C'}))$ ;

2. (ii) $\mathbb{E}[f(n)|n\in\Omega^\ast]\ge(1+2c')\delta$ .

Now our aim is to partition $\Omega^\ast$ into few arithmetic progressions, namely at most $X=7^dN^{1-\Omega(1/d^7)}$ progressions where $d=M'(\delta)$ . The number of elements of $\Omega^\ast$ in progressions of length at most $c' \delta |\Omega^\ast|/X$ is at most $c'\delta|\Omega^\ast|$ . Letting the union of short progressions be $\Omega'$ , we have that

\[\mathbb{E}[f(n)|n\in\Omega^\ast\setminus \Omega']\ge (1+c')\delta.\]

Therefore there is a progression of length longer than $c' \delta |\Omega^\ast|/X$ with density at least $(1+c')\delta$ . We have

\[c'\delta|\Omega^\ast|/X\ge N^{1/M(\delta)}\]

due to our assumption $N' \gt \exp\!(M(\delta))$ , assuming $C \gt 0$ is chosen sufficiently large. (This is where the dependence in the first item of Proposition 3·1 comes from.)

We may write

\[\Omega^\ast=[N']\cap\bigcap_{i\in I_1}\{x\colon\lVert\phi_i(x)-j_i/K\rVert_{\mathbb{R}/\mathbb{Z}}\le1/(2K)\}\cap\bigcap_{i\in I_2}\{x\colon\lVert\varphi_i(x)-j_i/K\rVert_{\mathbb{R}/\mathbb{Z}}\le1/(2K)\},\]

where $\phi_i$ are linear functions on $\mathbb{Z}/N\mathbb{Z}$ and $\varphi_i\colon T\to\mathbb{R}$ are locally cubic functions on

\[T\;:\!=\;[N']\cap\bigcap_{i\in I_1}\{x\colon\lVert\phi_i(x)-j_i/K\rVert_{\mathbb{R}/\mathbb{Z}}\le1/(2K)\}.\]

We additionally have $|I_1|,|I_2|,K\le M'(\delta)=:d$ . (The fact that $\Omega^\ast\subseteq[N']$ is evident from $\mathbb{E}[f(n)|n\in\Omega^\ast] \gt 0$ , recalling that f is extended by 0s in Proposition 3·6.)

Step 2. Partitioning T into progressions where the cubic phases are near-constant. Our goal will be to partition T into subprogressions on which each $\varphi_i$ for $i\in I_2$ is roughly constant $\mod 1$ . In fact, we will find a collection of progressions, say of the form $\{a+bn\colon n\in[L]\}$ , so that for $n\in[L]$ we have $\varphi_i(a+bn)=\kappa+P_i^{(a,b)}(n)\mod 1$ , where $P_i^{(a,b)}$ is a real polynomial of degree at most 3 with coefficients of size $O(L^{-4})$ .

We first partition

\[T=\bigsqcup_{j\in J_1}T_j,\]

where $T_j$ are progressions and $|J_1|\ll 2^{|I_1|}N^{1-1/(|I_1|+1)}$ . To do this, we may use [ 13 , proposition 6·3] (which is a simple application of Kronecker approximation, or Proposition 3·7 for $k=1$ ).

Since $T_j$ is a progression, we may write $T_j=\{a_j+d_j,\ldots,a_j+M_jd_j\}$ , where $M_j=|T_j|$ . Since each $\varphi_i$ is locally cubic on this progression, we may write

\[\varphi_i(a_j+d_jn)=\alpha_i^{(j)}n^3+\beta_i^{(j)}n^2+\gamma_i^{(j)}n+\delta_i^{(j)} \;\mathrm{mod}\;1\]

for $i\in I_2$ and $n\in[M_j]$ where $j\in J_1$ . Next, we perform three intermediate partitions of $T_j$ in order to iteratively “reduce the degree” of the function $\varphi_i$ until we see that it is roughly constant on our subprogressions.

First, choose $1\le r_j\le M_j^{1/3}$ so that

\[\max_{i\in I_2}\lVert\alpha_i^{(j)}r_j^3\rVert_{\mathbb{R}/\mathbb{Z}}\ll dM_j^{-c(3)/(3d^2)}.\]

Then we can partition $[M_j]$ into at most $M_j^{1-c(3)/(120d^2)}$ progressions of difference $r_j$ and length roughly $M_j^{c(3)/(120d^2)}$ . This corresponds to a partition

\[T_j=\bigsqcup_{k_3\in D_3}T_{j,k_3}\]

with $|D_3|\le M_j^{1-c(3)/(120d^2)}$ and $|T_{j,k_3}|\sim M_j^{c(3)/(120d^2)}$ . Furthermore, we have

\[T_{j,k_3}=\{a_j+(b_{j,k_3}+r_jt)d_j\colon t\in[|T_{j,k_3}|]\}\]

for some integer $b_{j,k_3}$ . We have

\begin{align*}\varphi_i(a_j+(b_{j,k_3}+r_jn)d_j)&=\alpha_i^{(j)}(b_{j,k_3}+r_jn)^3+\beta_i^{(j)}(b_{j,k_3}+r_jn)^2+\gamma_i^{(j)}(b_{j,k_3}+r_jn)+\delta_i^{(j)}\\&=\alpha_i^{(j,k_3)}n^2+\beta_i^{(j,k_3)}n+\gamma_i^{(j,k_3)}+\{\alpha_i^{(j)}r_j^3\}n^3 \;\mathrm{mod}\;1.\end{align*}

By construction, $\{\alpha_i^{(j)}r_j^3\}$ is close to $0\;\mathrm{mod}\;1$ (namely, it is $\ll dM_j^{-c(3)/(3d^2)}\ll|T_{j,k_3}|^{-30}$ ).

We can thus consider the quadratic function obtained by removing this part, and repeat the same process on each $T_{j,k_3}$ . We obtain iterative decompositions

\[T_{j,k_3}=\bigsqcup_{k_2\in D_2^{(k_3)}}T_{j,k_3,k_2},\quad T_{j,k_3,k_2}=\bigsqcup_{k_1\in D_1^{(k_3,k_2)}}T_{j,k_3,k_2,k_1},\]

where:

\begin{align*}|D_2^{(k_3)}|\le|T_{j,k_3}|^{1-c(2)/(120d^2)},\quad |D_1^{(k_3,k_2)}|\le|T_{j,k_3,k_2}|^{1-c(1)/(120d^2)};\\|T_{j,k_3,k_2}|\sim|T_{j,k_3}|^{c(2)/(120d^2)},\quad|T_{j,k_3,k_2,k_1}|\sim|T_{j,k_3,k_2}|^{c(1)/(120d^2)}.\end{align*}

Additionally, we will find that if $T_{j,k_3,k_2,k_1}=\{a+bn\colon n\in[|T_{j,k_3,k_2,k_1}|]\}$ then

\[\varphi_i(a+bn)=\alpha_i^{(j,k_3,k_2,k_1)}n^3+\beta_i^{(j,k_3,k_2,k_1)}n^2+\gamma_i^{(j,k_3,k_2,k_1)}n+\kappa_i^{(j,k_3,k_2,k_1)} \;\mathrm{mod}\;1,\]

where

\[|\alpha_i^{(j,k_3,k_2,k_1)}|+|\beta_i^{(j,k_3,k_2,k_1)}|+|\gamma_i^{(j,k_3,k_2,k_1)}|\ll|T_{j,k_3,k_2,k_1}|^{-4}.\]

The number of such $T_{j,k_3,k_2,k_1}$ partitioning $T_j$ is in total at most

\[M_j^{1-c(3)/(12d^2)}(M_j^{c(3)/(12d^2)})^{1-c(2)/(12d^2)}\big((M_j^{c(3)/(12d^2)})^{c(2)/(12d^2)}\big)^{1-c(1)/(12d^2)}\le M_j^{1-\Omega(1/d^6)}.\]

(In the rare cases of any j such that $M_j\le\exp\!(O(d^6))$ we may simply partition $M_j$ into progressions of length 1 appropriately.)

Finally, choosing appropriate value $p=\Omega(1/d^6)$ , the total number of subprogressions is

\[\sum_{j\in J_1}M_j^{1-p}\le|J_1|^p\bigg(\sum_{j\in J_1}M_j\bigg)^{1-p}\le|J_1|^pN^{1-p}\le N^{1-\Omega(1/d^7)}.\]

Step 3. Partitioning $\Omega^\ast$ into few progressions. From the previous part, we have a partition of T into at most $N^{1-\Omega(1/d^7)}$ progressions of various lengths so that for a progression $\{a+bn\colon n\in[L]\}$ that appears, we have

\[\varphi_i(a+bn)=\kappa+P_i^{(a,b)}(n) \;\mathrm{mod}\;1,\]

where $P_i^{(a,b)}$ is a real polynomial of degree at most 3 with coefficients of size $O(L^{-4})$ . Recalling $\Omega^\ast\subseteq T$ , we can now intersect these progressions with $\Omega^\ast$ . Note that $\Omega^\ast$ merely imposes a condition on each $\varphi_i$ for $i\in I_2$ . Since each $P_i^{(a,b)}(n)$ is at most degree 3, within the range $n\in[L]$ it hits the cutoffs for the condition for $\varphi_i$ at most $2\cdot 3=6$ times in the real numbers. Since $P_i^{(a,b)}(n)$ has small coefficients (so is small on $n\in[L]$ ), there is no rounding wraparound and the same holds for its image $\;\mathrm{mod}\;1$ over the domain $n\in[L]$ .

The upshot is that each progression is cut into at most 7 pieces by the condition on $\varphi_i$ defining $\Omega^\ast$ for each $i\in I_2$ . This gives at most $7^{|I_2|}\le 7^d$ total pieces.

Thus, we have a partition of $\Omega^\ast$ into at most $7^dN^{1-\Omega(1/d^7)}$ pieces, which combined with the argument above completes the proof.