A linear equation $E$ is said to be sparse if there is $c\gt 0$ so that every subset of $[n]$ of size $n^{1-c}$ contains a solution of $E$ in distinct integers. The problem of characterising the sparse equations, first raised by Ruzsa in the 90s, is one of the most important open problems in additive combinatorics. We say that $E$ in $k$ variables is abundant if every subset of $[n]$ of size $\varepsilon n$ contains at least $\text{poly}(\varepsilon )\cdot n^{k-1}$ solutions of $E$. It is clear that every abundant $E$ is sparse, and Girão, Hurley, Illingworth, and Michel asked if the converse implication also holds. In this note, we show that this is the case for every $E$ in four variables. We further discuss a generalisation of this problem which applies to all linear equations.

For a subset $A$ of an abelian group $G$, given its size $|A|$, its doubling $\kappa =|A+A|/|A|$, and a parameter $s$ which is small compared to $|A|$, we study the size of the largest sumset $A+A'$ that can be guaranteed for a subset $A'$ of $A$ of size at most $s$. We show that a subset $A'\subseteq A$ of size at most $s$ can be found so that $|A+A'| = \Omega (\!\min\! (\kappa ^{1/3},s)|A|)$. Thus, a sumset significantly larger than the Cauchy–Davenport bound can be guaranteed by a bounded size subset assuming that the doubling $\kappa$ is large. Building up on the same ideas, we resolve a conjecture of Bollobás, Leader and Tiba that for subsets $A,B$ of $\mathbb{F}_p$ of size at most $\alpha p$ for an appropriate constant $\alpha \gt 0$, one only needs three elements $b_1,b_2,b_3\in B$ to guarantee $|A+\{b_1,b_2,b_3\}|\ge |A|+|B|-1$. Allowing the use of larger subsets $A'$, we show that for sets $A$ of bounded doubling, one only needs a subset $A'$ with $o(|A|)$ elements to guarantee that $A+A'=A+A$. We also address another conjecture and a question raised by Bollobás, Leader and Tiba on high-dimensional analogues and sets whose sumset cannot be saturated by a bounded size subset.

In our paper, we study multiplicative properties of difference sets $A-A$ for large sets $A \subseteq {\mathbb {Z}}/q{\mathbb {Z}}$ in the case of composite q. We obtain a quantitative version of a result of A. Fish about the structure of the product sets $(A-A)(A-A)$. Also, we show that the multiplicative covering number of any difference set is always small.

For a nonempty set A of integers and an integer n, let $r_{A}(n)$ be the number of representations of n in the form $n=a+a'$, where $a\leqslant a'$ and $a, a'\in A$, and $d_{A}(n)$ be the number of representations of n in the form $n=a-a'$, where $a, a'\in A$. The binary support of a positive integer n is defined as the subset S(n) of nonnegative integers consisting of the exponents in the binary expansion of n, i.e., $n=\sum_{i\in S(n)} 2^i$, $S(-n)=-S(n)$ and $S(0)=\emptyset$. For real number x, let $A(-x,x)$ be the number of elements $a\in A$ with $-x\leqslant a\leqslant x$. The famous Erdős-Turán Conjecture states that if A is a set of positive integers such that $r_A(n)\geqslant 1$ for all sufficiently large n, then $\limsup_{n\rightarrow\infty}r_A(n)=\infty$. In 2004, Nešetřil and Serra initially introduced the notation of “bounded” property and confirmed the Erdős-Turán conjecture for a class of bounded bases. They also proved that, there exists a set A of integers satisfying $r_A(n)=1$ for all integers n and $|S(x)\bigcup S(y)|\leqslant 4|S(x+y)|$ for $x,y\in A$. On the other hand, Nathanson proved that there exists a set A of integers such that $r_A(n)=1$ for all integers n and $2\log x/\log 5+c_1\leqslant A(-x,x)\leqslant 2\log x/\log 3+c_2$ for all $x\geqslant 1$, where $c_1,c_2$ are absolute constants. In this paper, following these results, we prove that, there exists a set A of integers such that: $r_A(n)=1$ for all integers n and $d_A(n)=1$ for all positive integers n, $|S(x)\bigcup S(y)|\leqslant 4|S(x+y)|$ for $x,y\in A$ and $A(-x,x) \gt (4/\log 5)\log\log x+c$ for all $x\geqslant 1$, where c is an absolute constant. Furthermore, we also construct a family of arbitrarily spare such sets A.

Open conjectures state that, for every $x\in [0,1]$, the orbit $(x_n)_{n=1}^\infty $ of the mean-median recursion

$$ \begin{align*}x_{n+1}=(n+1)\cdot\operatorname{\mathrm{median}}(x_1,\ldots,x_{n})-(x_1+\cdots+x_n),\quad n\geqslant 3,\end{align*} $$

with initial data $(x_1,x_2,x_3)=(0,x,1)$, is eventually constant, and that its transit time and limit functions (of x) are unbounded and continuous, respectively. In this paper, we prove that for the slightly modified recursion

$$ \begin{align*}x_{n+1}=n\cdot\operatorname{\mathrm{median}}(x_1,\ldots,x_{n})-(x_1+\cdots+x_n),\quad n\geqslant 3,\end{align*} $$

first suggested by Akiyama, the transit time function is unbounded but the limit function is discontinuous.

A finite set of integers A tiles the integers by translations if $\mathbb {Z}$ can be covered by pairwise disjoint translated copies of A. Restricting attention to one tiling period, we have $A\oplus B=\mathbb {Z}_M$ for some $M\in \mathbb {N}$ and $B\subset \mathbb {Z}$. This can also be stated in terms of cyclotomic divisibility of the mask polynomials $A(X)$ and $B(X)$ associated with A and B.

In this article, we introduce a new approach to a systematic study of such tilings. Our main new tools are the box product, multiscale cuboids and saturating sets, developed through a combination of harmonic-analytic and combinatorial methods. We provide new criteria for tiling and cyclotomic divisibility in terms of these concepts. As an application, we can determine whether a set A containing certain configurations can tile a cyclic group $\mathbb {Z}_M$, or recover a tiling set based on partial information about it. We also develop tiling reductions where a given tiling can be replaced by one or more tilings with a simpler structure. The tools introduced here are crucial in our proof in [24] that all tilings of period $(pqr)^2$, where $p,q,r$ are distinct odd primes, satisfy a tiling condition proposed by Coven and Meyerowitz [2].

In 1946, Erdős and Niven proved that no two partial sums of the harmonic series can be equal. We present a generalisation of the Erdős–Niven theorem by showing that no two partial sums of the series $\sum _{k=0}^\infty {1}/{(a+bk)}$ can be equal, where a and b are positive integers. The proof of our result uses analytic and p-adic methods.

We demonstrate that every difference set in a finite Abelian group is equivalent to a certain ‘regular’ covering of the lattice $ A_n = \{ \boldsymbol {x} \in \mathbb {Z} ^{n+1} : \sum _{i} x_i = 0 \} $ with balls of radius $ 2 $ under the $ \ell _1 $ metric (or, equivalently, a covering of the integer lattice $ \mathbb {Z} ^n $ with balls of radius $ 1 $ under a slightly different metric). For planar difference sets, the covering is also a packing, and therefore a tiling, of $ A_n $ . This observation leads to a geometric reformulation of the prime power conjecture and of other statements involving Abelian difference sets.

We obtain a nontrivial upper bound for the multiplicative energy of any sufficiently large subset of a subvariety of a finite algebraic group. We also find some applications of our results to the growth of conjugates classes, estimates of exponential sums, and restriction phenomenon.

We improve the best known sum-product estimates over the reals. We prove that

\[\max(|A+A|,|A+A|)\geq |A|^{\frac{4}{3} + \frac{2}{1167} - o(1)}\,,\]

$A\subset \mathbb {R}$

Furthermore,

\[|AA+AA|\geq |A|^{\frac{127}{80} - o(1)}\,.\]

\[|A+A|\geq |A|^{\frac{30}{19}-o(1)}\,.\]

We offer an alternative proof of a result of Conlon, Fox, Sudakov and Zhao [CFSZ20] on solving translation-invariant linear equations in dense Sidon sets. Our proof generalises to equations in more than five variables and yields effective bounds.

Fix positive integers k and n with $k \leq n$ . Numbers $x_0, x_1, x_2, \ldots , x_{n - 1}$ , each equal to $\pm {1}$ , are cyclically arranged (so that $x_0$ follows $x_{n - 1}$ ) in that order. The problem is to find the product $P = x_0x_1 \cdots x_{n - 1}$ of all n numbers by asking the smallest number of questions of the type $Q_i$ : what is $x_ix_{i + 1}x_{i + 2} \cdots x_{i+ k -1}$ ? (where all the subscripts are read modulo n). This paper studies the problem and some of its generalisations.

Let k and l be positive integers satisfying $k \ge 2, l \ge 1$ . A set $\mathcal {A}$ of positive integers is an asymptotic basis of order k if every large enough positive integer can be represented as the sum of k terms from $\mathcal {A}$ . About 35 years ago, P. Erdős asked: does there exist an asymptotic basis of order k where all the subset sums with at most l terms are pairwise distinct with the exception of a finite number of cases as long as $l \le k - 1$ ? We use probabilistic tools to prove the existence of an asymptotic basis of order $2k+1$ for which all the sums of at most k elements are pairwise distinct except for ‘small’ numbers.

We prove that if $A \subseteq [X,\,2X]$ and $B \subseteq [Y,\,2Y]$ are sets of integers such that gcd (a, b) ⩾ D for at least δ|A||B| pairs (a, b) ε A × B then $|A||B|{ \ll _{\rm{\varepsilon }}}{\delta ^{ - 2 - \varepsilon }}XY/{D^2}$ . This is a new result even when δ = 1. The proof uses ideas of Koukoulopoulos and Maynard and some additional combinatorial arguments.

A set of integers is primitive if it does not contain an element dividing another. Let f(n) denote the number of maximum-size primitive subsets of {1,…,2n}. We prove that the limit α = limn→∞f(n)1/n exists. Furthermore, we present an algorithm approximating α with (1 + ε) multiplicative error in N(ε) steps, showing in particular that α ≈ 1.318. Our algorithm can be adapted to estimate the number of all primitive sets in {1,…,n} as well.

We address another related problem of Cameron and Erdős. They showed that the number of sets containing pairwise coprime integers in {1,…n} is between ${2^{\pi (n)}} \cdot {e^{(1/2 + o(1))\sqrt n }}$ and ${2^{\pi (n)}} \cdot {e^{(2 + o(1))\sqrt n }}$. We show that neither of these bounds is tight: there are in fact ${2^{\pi (n)}} \cdot {e^{(1 + o(1))\sqrt n }}$ such sets.

Let $n$ be a positive integer and $a$ an integer prime to $n$. Multiplication by $a$ induces a permutation over $\mathbb{Z}/n\mathbb{Z}=\{\overline{0},\overline{1},\ldots ,\overline{n-1}\}$. Lerch’s theorem gives the sign of this permutation. We explore some applications of Lerch’s result to permutation problems involving quadratic residues modulo $p$ and confirm some conjectures posed by Sun [‘Quadratic residues and related permutations and identities’, Preprint, 2018, arXiv:1809.07766]. We also study permutations involving arbitrary $k$th power residues modulo $p$ and primitive roots modulo a power of $p$.

We prove that, given a positive integer $m$, there is a sequence $\{n_{i}\}_{i=1}^{k}$ of positive integers such that

$$\begin{eqnarray}m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots +\frac{1}{n_{k}}\end{eqnarray}$$

$\{1/n_{i}\}_{i=1}^{k}$

Suppose that N is 2-coloured. Then there are infinitely many monochromatic solutions to $x+y=z^{2}$. On the other hand, there is a 3-colouring of N with only finitely many monochromatic solutions to this equation.

In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding three-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid three-term geometric progressions. When unique factorization fails, or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets ‘greedily’, a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometric-progression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.

For $n= 1, 2, 3, \ldots $ let ${S}_{n} $ be the sum of the first $n$ primes. We mainly show that the sequence ${a}_{n} = \sqrt[n]{{S}_{n} / n}~(n= 1, 2, 3, \ldots )$ is strictly decreasing, and moreover the sequence ${a}_{n+ 1} / {a}_{n} ~(n= 10, 11, \ldots )$ is strictly increasing. We also formulate similar conjectures involving twin primes or partitions of integers.