Multiplicative dependence in linear recurrence sequences

Abstract

For a wide class of integer linear recurrence sequences $(u(n))_{n=1}^\infty $, we give an upper bound on the number of s-tuples $\left (n_1, \ldots , n_s\right ) \in \left ({\mathbb Z}\cap [M+1,M+ N]\right )^s$ such that the corresponding elements $u(n_1), \ldots , u(n_s)$ in the sequence are multiplicatively dependent.

1 Introduction

1.1 Motivation and set-up

Let $\mathbf {u} = \left (u(n)\right )_{n=1}^\infty $ be an integer linear recurrence sequence of order $d \ge 1$ , that is, a sequence of integers satisfying a relation of the form

$$\begin{align*}u(n+d) = c_{d-1} u(n+d-1) +\cdots + c_{0} u(n) , \qquad n =1,2, \ldots, \end{align*}$$

and not satisfying any shorter relation. In this case

$$\begin{align*}f(X) = X^d - c_{d-1} X^{n+d-1} -\cdots - c_{0} \in {\mathbb Z}[X] \end{align*}$$

is called the characteristic polynomial of $\mathbf {u}$ .

Recently there have been several works [3–6, 9–11, 13] investigating multiplicative relations of the form

(1.1) $$ \begin{align} u(n_1)^{k_1} \ldots u(n_s)^{k_s} = 1. \end{align} $$

However, these papers consider certain special cases. The works [6, 11, 13] are limited to the case of binary (that is, of order $d =2$ ) linear recurrence sequences and also assume that the exponents $k_1, \ldots , k_s$ are fixed nonzero integers, while the papers [3, 4, 9, 10] concern specific sequences. Under these restrictions, the mentioned papers contain several finiteness results. Finally, the recent work [5] concerns linear recurrence sequences of arbitrary order—however, under a rather restrictive condition on the coefficients $c_i$ defining the generating relation.

Here we are interested in the case of general sequences of arbitrary order $d \ge 2$ and also we do not fix the exponents $k_1, \ldots , k_s$ . Thus, we study s-tuples $\left (u(n_1), \ldots , u(n_s)\right )$ , which are multiplicatively dependent (m.d.), where, as usual, we say that the nonzero complex numbers $\gamma _1,\ldots ,\gamma _s$ are m.d. if there exist integers $k_1,\ldots ,k_s$ , not all zero, such that

$$\begin{align*}\gamma_1^{k_1}\ldots \gamma_s^{k_s}=1. \end{align*}$$

However, instead of finiteness results, we give an upper bound on the density of such s-tuples.

More precisely, for $M\ge 0$ and $N\ge 1$ , we are interested in the following quantity

$$ \begin{align*} \mathsf{M}_s(M,N)=\sharp\{(n_1&,\ldots,n_s)\in\left({\mathbb Z}\cap [M+1,M+N]\right)^s:\\ &\qquad\qquad\qquad\qquad ~u(n_1),\ldots,u(n_s)\ \text{are} \ \textsf{m.d.}\}. \end{align*} $$

To estimate $\mathsf {M}_s(M,N)$ , we also study

$$ \begin{align*} \mathsf{M}_s^*(M,N)=\sharp\{(n_1&,\ldots,n_s)\in\left({\mathbb Z}\cap [M+1,M+N]\right)^s: \\ & ~u(n_1),\ldots,u(n_s)\ \text{are } \ \textsf{m.d.}\ \text{of maximal rank}\}, \end{align*} $$

where the maximality of the rank for m.d. of $(u(n_1),\ldots ,u(n_s))$ means that no sub-tuple is m.d. In particular, this implies that if one has a m.d. (1.1) of maximal rank, then $k_1\cdots k_s \ne 0$ .

We can then estimate $\mathsf {M}_s(M,N)$ via the inequality

(1.2) $$ \begin{align} \mathsf{M}_s(M,N) \le \sum_{t=1}^s \binom{s}{t} \mathsf{M}_t^*(M,N) N^{s-t}. \end{align} $$

1.2 Notation

We recall that the notations $U = O(V)$ , $U \ll V$ , and $ V\gg U$ are equivalent to $|U|\leqslant c V$ for some positive constant c, which throughout this work, may depend only on the integer parameter s and the sequence $\mathbf {u}$ .

It is convenient to denote by $\log _{k} x$ the k-fold iterated logarithm, that is, for $x\ge 1$ we set

$$\begin{align*}\log_1 x = \log x \qquad\mbox{and}\qquad \log_k = \log_{k-1} \max\{\log x, 2\}, \quad k =2, 3, \ldots. \end{align*}$$

1.3 Main results

We say that the sequence $\mathbf {u}$ is non-degenerate if there are no roots of unity among the ratios of distinct roots of f. We say that the sequence $\mathbf {u}$ has a dominant root, if its characteristic polynomial f has a root $\lambda $ with

$$\begin{align*}|\lambda|> \max\{|\mu|:~ f(\mu) = 0, \ \mu \ne \lambda\}. \end{align*}$$

Furthermore, we say that $\mathbf {u}$ is simple if f has no multiple roots.

Theorem 1.1 Let $\mathbf {u}$ be a simple non-degenerate sequence of order $d \ge 2$ . For any fixed $s \ge 1$ , uniformly over $M\ge 0$ , we have

$$\begin{align*}\mathsf{M}_s^*(M,N)\le N^{s\left(1-1/(4d-3)\right)+o(1)}. \end{align*}$$

Analyzing the proof of Theorem 1.1, one can see that for $M=0$ we can drop $o(1)$ in the bound.

Remark 1.2 Considering s-tuples with $n_1=n_2$ we see that

(1.3) $$ \begin{align} \mathsf{M}_s(M,N) \ge N^{s-1}. \end{align} $$

Therefore, it is impossible to derive a bound of the same type as in Theorem 1.1 for $\mathsf {M}_s(M,N)$ .

When M is (exponentially) large compared to N, we get the following bound, which improves Theorem 1.1 for $s<4d-3$ .

Theorem 1.3 Let $\mathbf {u}$ be a simple non-degenerate sequence of order $d \ge 2$ with a dominant root and let

$$\begin{align*}M\ge \exp(N \log_3N/\log_2N). \end{align*}$$

Then, for any fixed $s \ge 1$ , uniformly over M, we have

$$\begin{align*}\mathsf{M}_s^*(M,N)\le N^{s-1+o(1)}. \end{align*}$$

Remark 1.4 The condition on M in Theorem 1.3 is chosen to achieve the strongest possible bound. Examining its proof one can see that for $s < 4d-3$ one can also improve Theorem 1.1 for $M\ge \exp (N^\eta )$ with any $\eta> s/(4d-3)$ (but only for sequences with a dominant root).

From the definition of m.d. of maximal rank, we have $\mathsf {M}_1^*(M,N) =O(1)$ , see [1, Lemma 2.1]. Hence, we see from (1.2) that in applying Theorem 1.1 to bounding $\mathsf {M}_s(M,N)$ the case of $s = 2$ becomes the bottleneck. Thus, we now investigate this case separately.

Theorem 1.5 Let $\mathbf {u}$ be a simple non-degenerate sequence of order $d \ge 2$ with an irreducible characteristic polynomial having a dominant root. Uniformly over $M\ge 0$ , we have

$$\begin{align*}\mathsf{M}_2^*(M,N) = N+ O(1). \end{align*}$$

Since, as we have mentioned, $\mathsf {M}_1^*(M,N)=O(1)$ , the bounds of Theorems 1.1 and 1.5 inserted in (1.2) imply that if $\mathbf {u}$ is a simple non-degenerate sequence of order $d \ge 2$ with an irreducible characteristic polynomial having a dominant root then

(1.4) $$ \begin{align} \mathsf{M}_s(M,N) \ll N^{s - 3/(4d-3)+o(1)}, \end{align} $$

where the bottleneck comes from the bound on $\mathsf {M}_3^*(M,N)$ . In fact in this bound the condition of irreducibility can be dropped, see Remark 3.2 below.

If $M\ge \exp (N \log _3N/\log _2N)$ , then using instead Theorem 1.3, one obtains the upper bound

$$\begin{align*}\mathsf{M}_s(M,N) \ll N^{s - 1+o(1)}, \end{align*}$$

which matches the trivial lower bound (1.3).

Remark 1.6 Analyzing the proofs, one can easily see that the above results extend without any changes to m.d. in s-tuples $\left (u_1(n_1), \ldots , u_s(n_s)\right )$ , of s (not necessary distinct) linear recurrence sequences.

2 Preliminaries

2.1 Arithmetic properties of linear recurrence sequences

In this section, we collect various results about the arithmetic properties of a linear recurrence sequence that we need for our main results. These include:

• • a lower bound of square-free parts of elements in $\mathbf {u}$ ,

• • a bound for the number of elements in $\mathbf {u}$ that are S-units,

• • various results on congruences with elements in $\mathbf {u}$ ,

• • a result on the finiteness of perfect powers in $\mathbf {u}$ .

Some of these are obtained under the condition that $\mathbf {u}$ has a dominant root.

We start with a lower bound of Stewart [17, Theorem 1] on the square-free part of elements in a linear recurrence.

For any integer m, we define $\mathrm {rad}(m)$ to be the largest square-free factor of m.

Lemma 2.1 Let $\mathbf {u}$ be a simple non-degenerate sequence of order $d \ge 2$ with a dominant root. Then there exist constants $C_1$ and $C_2$ , which are effectively computable only in terms of $\mathbf {u}$ , such that if $n\ge C_2$ , then

$$\begin{align*}\mathrm{rad}(u(n))>n^{C_1(\log_2 n)/\log_3 n}. \end{align*}$$

We also need the following upper bound from [15, Theorem 1 and Corollary] on the number of terms of $\mathbf {u}$ composed out of primes from a given set. We note that the condition of the exponential growth of the terms of $\mathbf {u}$ , assumed in [15], is now known to hold for non-degenerate recurrence sequences, see [8, 14]. Hence, we have the following result.

Lemma 2.2 Let $\mathbf {u}$ be a non-degenerate sequence of order $d \ge 2$ and let S be an arbitrary set of r primes. Then, for $M\ge 0$ , the number $A(S;M,N)$ of terms $u(M+1),\ldots , u(M+N)$ , composed exclusively of primes from S, satisfies

$$\begin{align*}A(S;M,N) \ll \begin{cases} rNM^{-1}\log(N+M)& \text{for } M \ge 1,\\ r(\log N)^2& \text{for } M =0. \end{cases} \end{align*}$$

We now present two results regarding solutions to certain congruences with elements in a linear recurrence sequence. We start with a result, which follows from [15, Lemmas 2 and 3].

Lemma 2.3 Let $\mathbf {u}$ be a non-degenerate sequence of order $d \ge 2$ and let $m\ge 1$ be an integer. Then we have

$$\begin{align*}\sharp\{n \in {\mathbb Z}\cap [M+1,M+N]:~u(n)\equiv 0\quad \pmod{m}\}\ll N/\log m+1. \end{align*}$$

The second bound that we need holds modulo primes and follows from [2, Lemma 6]. In [2], it is formulated only for the interval $[1,N]$ , however the result is uniform with respect to the sequence $\mathbf {u}$ and hence it holds uniformly with respect to M, too.

Let $\overline {{\mathbb F}}_p$ be the algebraic closure of the finite field ${\mathbb F}_p$ of p elements.

Lemma 2.4 Let $\mathbf {u}$ be a simple sequence of order $d \ge 2$ and for a prime p let $\lambda _1, \ldots , \lambda _d$ be the roots of the characteristic polynomial of $\mathbf {u}$ in $ \overline {{\mathbb F}}_p$ . We set $\varrho _p = 1$ if at least one root $\lambda _1, \ldots , \lambda _d$ is zero and set

$$\begin{align*}\varrho_p=\min\limits_{1\leq i< j\leq d} r_{ij}, \end{align*}$$

where $r_{ij}$ is the multiplicative order of $\lambda _i/\lambda _j$ in $\overline {{\mathbb F}}_p$ , otherwise. Then for any integers $M\ge 0$ and $N\ge 1$ , we have

$$\begin{align*}\sharp\{n \in {\mathbb Z}\cap [M+1,M+N]:~u(n)\equiv 0\quad \pmod{p}\}\ll N(N^{-1}+\varrho_p^{-1})^{1/(d-1)}. \end{align*}$$

The following result is certainly well-known and is based on classical ideas of Hooley [12], however for completeness we present a short proof.

Lemma 2.5 For $R\ge 2$ we consider the set

$$\begin{align*}{\mathcal W}(R)=\{p\ \text{prime}:~\varrho_p\leq R\}. \end{align*}$$

Then $\sharp {\mathcal W}(R) \ll R^2/\log R$ .

Proof Write $\lambda _1,\dots ,\lambda _q$ for the distinct roots of the characteristic polynomial of $\mathbf {u}$ .

For $R \ge 2$ , let

$$\begin{align*}Q(R) = \prod_{\rho \leq R} \prod_{1\leq i<j\leq q} {\mathrm{Nm}}_{K/{\mathbb Q}}(\lambda_i^\rho-\lambda_j^\rho), \end{align*}$$

where ${\mathrm {Nm}}_{K/{\mathbb Q}}$ is the norm from the splitting field K of f to ${\mathbb Q}$ . Note that $Q(R)\neq 0$ because $\lambda _i/\lambda _j$ is not a root of unity and since $\lambda _i$ and $\lambda _j$ are algebraic integers we also have $ Q(R) \in {\mathbb Z}$ .

Clearly, for any prime p which does not divide the constant coefficient of the characteristic polynomial of $\mathbf {u}$ and with $\varrho _p\leq R$ , we have $p \mid Q(R)$ , hence

$$\begin{align*}\sharp {\mathcal W}(R)\le \omega(Q(R)) +O(1), \end{align*}$$

where $\omega (k)$ is the number of prime divisors of an integer $k \ge 1$ . As clearly $\omega (k)! \le k$ , by the Stirling formula we get

$$\begin{align*}\sharp {\mathcal W}(R) \ll \frac{\log Q(R)}{\log \log Q(R)}. \end{align*}$$

Since obviously $\log Q(R) \ll R^2$ , the result follows.

Finally, we need a result on the finiteness of perfect powers in linear recurrence sequences with a dominant root. The most general and convenient form for us, which is built on several previous results in this direction, is given by of Bugeaud and Kaneko [7, Theorem 1.1].

Lemma 2.6 Let $\mathbf {u}$ be a simple non-degenerate sequence of order $d \ge 2$ with an irreducible characteristic polynomial having a dominant root. Then the equation ${u(n) = m^k}$ has only finitely many solutions in integer $k\ge 2$ , $m \ne 0$ , $n \ge 1$ .

2.2 Vertex covers

We need the following graph-theoretic result.

Lemma 2.7 Let G be a graph with vertex set ${\mathcal V}$ , having no isolated vertex. Put $\ell =\sharp {\mathcal V}$ . Then there exists $ {\mathcal V}_1\subseteq {\mathcal V}$ with $\sharp {\mathcal V}_1\leq \ell /2$ such that for any $v_2\in {\mathcal V}_2= {\mathcal V}\setminus {\mathcal V}_1$ there exists a vertex $v_1\in {\mathcal V}_1$ which is a neighbor of $v_2$ .

Proof The statement must be well-known, but we give a simple proof. If $\widetilde G$ is a graph (without isolated vertices) obtained from G by omitting some edges, and the statement is valid for $\widetilde G$ , then the statement is obviously valid for G. Let $\widetilde G$ be a forest graph (that is, a graph without cycles) obtained from G by omitting some edges, such that the number of connected components of G and $\widetilde G$ are the same. Then $\widetilde G$ is a bipartite graph, so the statement is clearly valid for it. Hence the result follows.

3 Proofs

3.1 Proof of Theorem 1.1

Suppose that for some $n_1,\ldots ,n_s \in [M+1,M+N]$ the terms $u(n_1),\ldots ,u(n_s)$ are m.d. of maximal rank, that is, we have (1.1) with some nonzero integers $k_1,\ldots ,k_s$ .

Choose a positive real number $R\ge 2$ to be specified later, and let ${\mathcal W}(R)$ be as in Lemma 2.5.

Write t for the number of indices $i =1, \ldots , s$ for which $u(n_i)$ has a prime divisor $p_i\notin {\mathcal W}(R)$ , and let $r = s-t$ for the number of indices i with $u(n_i)$ having all prime divisors in ${\mathcal W}(R)$ . Without loss of generality, we may assume that the corresponding integers are $n_1,\ldots ,n_t$ , and $n_{t+1},\ldots ,n_s$ , respectively.

By Lemmas 2.2 and 2.5, for $M\ge 1$ , the number $K_1$ of such r-tuples $\left (n_{t+1},\ldots ,n_s\right ) \in [M+1,M+N]^r$ satisfies

(3.1) $$ \begin{align} K_1\ll \left(\frac{R^2N\log (N+M)}{M\log R}\right)^r. \end{align} $$

If $M=0$ , then we have the bound

(3.2) $$ \begin{align} K_1\ll \left(\frac{R^2(\log N)^2}{\log R}\right)^r. \end{align} $$

We assume that such an r-tuple $\left (n_{t+1},\ldots ,n_s\right )$ is fixed.

Consider the t-tuples $\left (n_1,\ldots ,n_t\right )\in [M+1,M+N]^t$ . Recall that for any $1\leq i\leq t$ , there is a prime $p_i\notin {\mathcal W}(R)$ such that $p_i\mid u(n_i)$ .

Define the graph ${\mathcal G}$ whose vertices are $u(n_1),\ldots ,u(n_t)$ , and connect the vertices $u(n_i)$ and $u(n_j)$ precisely when $\gcd (u(n_i),u(n_j))$ has a prime divisor outside ${\mathcal W}(R)$ . Observe that as $u(n_1),\ldots ,u(n_s)$ are m.d. of maximal rank, ${\mathcal G}$ has no isolated vertex. Thus, by Lemma 2.7, there exists a subset ${\mathcal I}$ of $\{1,\ldots , t\}$ with

(3.3) $$ \begin{align} m=\sharp {\mathcal I}\leq \left\lfloor t/2\right\rfloor \end{align} $$

such that for any j with

$$\begin{align*}j\in \{n_1,\ldots,n_t\}\setminus {\mathcal I} \end{align*}$$

the vertex $u(n_j)$ is connected with some $u(n_i)$ in ${\mathcal G}$ , for some $i\in {\mathcal I}$ .

Without loss of generality, we may assume that ${\mathcal I}=\{1,\ldots , m\}$ . Trivially, the number $K_2$ of such m-tuples $(n_1,\ldots ,n_m)\in [M+1,M+N]^m$ satisfies

(3.4) $$ \begin{align} K_2\ll N^m. \end{align} $$

We now fix such an m-tuple. For $\ell = t-m$ , we now count the number $K_3$ of the remaining $\ell $ -tuples $(n_{m+1},\ldots ,n_t) \in [M+1,M+N]^\ell $ . Since each $u(n_j)$ with ${m+1\leq j\leq t}$ has a common prime factor $p\notin {\mathcal W}(R)$ with $u(n_i)$ for some $1\leq i\leq m$ , by Lemma 2.4 we obtain that $n_j$ comes from a set ${\mathcal N}$ of cardinality

$$\begin{align*}\sharp {\mathcal N} \ll N\left(N^{-1}+\varrho_p^{-1}\right)^{1/(d-1)}\leq N\left(N^{-1}+R^{-1}\right)^{1/(d-1)}. \end{align*}$$

Thus we obtain

(3.5) $$ \begin{align} K_3 \le \left(\sharp {\mathcal N}\right)^\ell \ll \left(N\left(N^{-1}+R^{-1}\right)^{1/(d-1)}\right)^{t-m}. \end{align} $$

We consider now two cases based on $M\le N\log N$ or $M> N\log N$ .

If $M\le N\log N$ , then

$$\begin{align*}\mathsf{M}_s^*(M,N)\le \mathsf{M}_s^*(0,2N\log N), \end{align*}$$

therefore we reduce to counting s-tuples in the interval $[0,2N\log N]^s$ .

Putting together the bounds (3.2), (3.4), and (3.5) (with N replaced by $2N\log N$ ), for some non-negative integer $t \le s$ and $r = s-t$ , we obtain

(3.6) $$ \begin{align} \begin{aligned} \mathsf{M}_s^*(M,N) &\le K_1K_2K_3\\ &\ll \left(R^2(\log N+\log\log N)^2/\log R\right)^r\left(N \log N\right)^m \\ & \qquad\qquad \qquad \left(N\log N\left(N^{-1}+R^{-1}\right)^{1/(d-1)}\right)^{t-m}\\ & \le N^{t+o(1)} R^{2r} \left(\left(N^{-1}+R^{-1}\right)^{1/(d-1)}\right)^{t/2}, \end{aligned} \end{align} $$

where the last inequality comes from (3.3).

Letting $R=N^\eta $ with some $0<\eta <1/2$ , we obtain

$$\begin{align*}\mathsf{M}_s^*(M,N)\ll N^{t + 2\eta r - \eta t/(2(d-1)) + o(1)} = N^{2\eta s + (1-2 \eta) t - \eta t/(2(d-1)) + o(1)}. \end{align*}$$

Writing $t= z s$ (and noting that $0\leq z\leq 1$ ), the exponent of the last term above (omitting the expression $o(1)$ ) is given by

$$\begin{align*}f_\eta(z) = \frac{s}{2(d-1)}\left((2d-4d\eta+3\eta-2)z+4\eta(d-1)\right). \end{align*}$$

So taking

$$\begin{align*}\eta = \frac{2(d-1)}{4d-3}, \end{align*}$$

(to make $f_\eta (z)$ a constant), we obtain

$$\begin{align*}\mathsf{M}_s^*(M,N)\ll N^{2\eta s + o(1)} = N^{s - s/(4d-3) + o(1)}, \end{align*}$$

which concludes this case.

If $M>N\log N$ , then the bound (3.1) becomes

$$\begin{align*}K_1\ll (R^2/\log R)^r. \end{align*}$$

Putting this together with (3.4) and (3.5), we obtain (3.6) without the $(\log N)^2$ factor, that is,

$$ \begin{align*} \mathsf{M}_s^*(M,N)& \le K_1K_2K_3\\ & \ll (R^2/\log R)^rN^m \left(N\left(N^{-1}+R^{-1}\right)^{1/(d-1)}\right)^{t-m}\\ & \ll N^{t} R^{2r} \left(\left(N^{-1}+R^{-1}\right)^{1/(d-1)}\right)^{t/2}. \end{align*} $$

Using the same discussion and choice of $\eta $ as above, we conclude the proof.

Remark 3.1 Clearly in (3.6), we can replace $t/2$ with $\left \lceil t/2\right \rceil $ but this does not change the optimal choice of $\eta $ and thus the final bound.

3.2 Proof of Theorem 1.3

Let $(n_1,\ldots ,n_s)\in [M+1,M+N]^s$ such that $u(n_1),\ldots ,u(n_s)$ is m.d. of maximal rank, which implies that there exist integers $k_i\ne 0$ , $i=1,\ldots ,s$ , such that (1.1) holds. We can rewrite this relation as

(3.7) $$ \begin{align} \prod_{i \in {\mathcal I}}u(n_i)^{k_i} = \prod_{j\in {\mathcal J}} u(n_j)^{k_j},\quad k_i,k_j>0, \end{align} $$

where ${\mathcal I}\cup {\mathcal J}=\{1,\ldots ,s\}$ , ${\mathcal I}\neq \emptyset $ , ${\mathcal J}\neq \emptyset $ , ${\mathcal I}\cap {\mathcal J}=\emptyset $ . Let $I=\sharp {\mathcal I}$ and $J=\sharp {\mathcal J}$ , and thus, ${I+J=s}$ .

Fix one of $2^s-2$ possible choices of the sets ${\mathcal I}$ and ${\mathcal J}$ as above. Fix $n_i$ , $i\in {\mathcal I}$ , trivially in $O(N^I)$ ways. Then, the square-free part $\mathrm {rad}(u(n_i))$ of $u(n_i)$ is fixed for each $i\in {\mathcal I}$ .

We may also assume that $n_i\ge C_2$ , $i\in {\mathcal I}$ , with $C_2$ as in Lemma 2.1, since this condition is violated only for $O(N^{s-1})$ choices of $(n_1, \ldots , n_s)$ , which is admissible. By Lemma 2.1, for $n_i\in [M+1,M+N]$ , one has

(3.8) $$ \begin{align} \mathrm{rad}(u(n_i))>n_i^{c(\log_2 n_i)/\log_3 n_i}\gg M^{c(\log_2 M)/\log_3(M+N)}. \end{align} $$

For $i\in {\mathcal I}$ , from (3.7) we see

$$\begin{align*}\mathrm{rad}(u(n_i)) \mid \prod_{j\in {\mathcal J}} u(n_j). \end{align*}$$

This implies that there is a factorization $\mathrm {rad}(u(n_i))=d_1\cdots d_J$ such that for each positive integer $d_\ell $ there exists $j\in {\mathcal J}$ such that $d_\ell \mid u(n_{j})$ . Let $\ell $ , $1 \le \ell \le J$ , be such that $d_\ell \ge \mathrm {rad}(u(n_i))^{1/J}$ , and

(3.9) $$ \begin{align} u(n_j)\equiv 0 \pmod{d_\ell}. \end{align} $$

From (3.8), we have

(3.10) $$ \begin{align} d_\ell>M^{c_0(\log_2 M)/\log_3(M+N)} \end{align} $$

with $c_0 = c/J \ge c/s$ .

Using now Lemma 2.3, the inequality (3.10) and the fact that $J\le s$ , the number of $n_j\in [M+1,M+N]$ satisfying the congruence (3.9) is

$$\begin{align*}O\left(N/\log d_{\ell} +1\right)=O\left( N\frac{\log_3(M+N)}{\log M \log_2 M}+1\right). \end{align*}$$

Therefore, using the trivial bound $N^{J-1}$ for the number of the remaining choices of $n_j$ with $j\in {\mathcal J}$ , we obtain that the total number of $n_j\in [M+1,M+N]$ , $j\in {\mathcal J}$ , is

$$\begin{align*}O\left(N^J\frac{\log_3(M+N)}{\log M \log_2 M}+N^{J-1}\right). \end{align*}$$

Thus we obtain that

$$\begin{align*}\mathsf{M}_s^*(M,N)\ll N^s \frac{\log_3(M+N)}{\log M \log_2 M}+ N^{s-1}. \end{align*}$$

Choosing $M\ge \exp (N \log _3N/\log _2N)$ , we conclude the proof.

3.3 Proof of Theorem 1.5

Clearly for $s=2$ we have to count integers $M+1\le m, n \le M+N$ , with

(3.11) $$ \begin{align} u(m)^a = u(n)^b \end{align} $$

for some positive integers a and b, where without loss of generality we can assume that $\gcd (a,b) = 1$ . We also notice that since the relation (3.11) is of maximal rank, neither $u(m)=\pm 1$ nor $u(n)=\pm 1$ holds.

Since $\mathbf {u}$ has a dominant root, $|u(n)|$ grows monotonically with n, provided that n is large enough. Hence there are $N+O(1)$ solutions $(m,n)\in [M+1,M+N]^2$ with $a=b=1$ .

Now we count pairs $(m,n)$ for which (3.11) holds with some $(a,b) \ne (1,1)$ .

We observe that if $a>1$ then $u(n)$ is the ath power and by Lemma 2.6 there are $O(1)$ such values of n. For $b> 1$ the argument also applies to m. Hence the total contribution from such solutions, over all $a, b> 1$ , is $O(1)$ .

If $a> 1$ and $b = 1$ , then again we see that there are $O(1)$ such values of n. From this we easily derive that $a= O(1)$ , and hence we obtain $O(1)$ possible values for m. So the contribution of such solutions to (3.11) is also O(1) only.

The case of $a=1$ and $b>1$ is completely analogous, which concludes the proof.

Remark 3.2 We note that without the irreducibility condition of the characteristic polynomial, that is, only under the condition of having a dominant root, we have boundedness of k in Lemma 2.6, see the discussion in [7, Section 1]. Thus, the above proof shows that in this case we have a version of Theorem 1.5 in the form $\mathsf {M}_2^*(M,N) \ll N$ and thus (1.4) holds only under this assumption.

4 Possible applications of our approach

Our approach works for many other integer sequences $\left (a(n)\right )_{n=1}^\infty $ , provided the following information is available:

1. (i) there are good bounds on the number of solutions to congruences $a(n) \equiv 0 \pmod q$ , $1 \le n \le N$ , in a broad range of positive integers q (or even just prime $q=p$ ) and N;

2. (ii) there are good bounds (or known finiteness) on the number of perfect powers among $a(n)$ , $1 \le n \le N$ .

For example, using results of [16], coupled with the finiteness result of Lemma 2.6, one can estimate the number of multiplicatively dependent s-tuples from values of linear recurrence sequences at polynomial values of the argument $\left (u\left (F(n)\right )\right )_{n=1}^\infty $ , where $F\in {\mathbb Z}[X]$ .