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 [Reference Batte, Ddamulira, Kasozi and Luca3–Reference Bérczes and Ziegler6, Reference Gómez, Gómez and Luca9–Reference Gómez Ruiz and Luca11, Reference Luca and Ziegler13] investigating multiplicative relations of the form
$$ \begin{align} u(n_1)^{k_1} \ldots u(n_s)^{k_s} = 1. \end{align} $$
However, these papers consider certain special cases. The works [Reference Bérczes and Ziegler6, Reference Gómez Ruiz and Luca11, Reference Luca and Ziegler13] 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 [Reference Batte, Ddamulira, Kasozi and Luca3, Reference Behera and Ray4, Reference Gómez, Gómez and Luca9, Reference Gómez Ruiz and Luca10] concern specific sequences. Under these restrictions, the mentioned papers contain several finiteness results. Finally, the recent work [Reference Behera and Ray5] 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
$$ \begin{align} \mathsf{M}_s(M,N) \le \sum_{t=1}^s \binom{s}{t} \mathsf{M}_t^*(M,N) N^{s-t}. \end{align} $$
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
$$ \begin{align} \mathsf{M}_s(M,N) \ge N^{s-1}. \end{align} $$
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 [Reference Amoroso and Viada1, 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
$$ \begin{align} \mathsf{M}_s(M,N) \ll N^{s - 3/(4d-3)+o(1)}, \end{align} $$
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 [Reference Stewart17, 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 [Reference Shparlinski15, 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 [Reference Shparlinski15], is now known to hold for non-degenerate recurrence sequences, see [Reference Evertse8, Reference van der Poorten and Schlickewei14]. 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 [Reference Shparlinski15, 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 [Reference Banks, Friedlander, Konyagin and Shparlinski2, Lemma 6]. In [Reference Banks, Friedlander, Konyagin and Shparlinski2], 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 [Reference Hooley12], 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 [Reference Bugeaud and Kaneko7, 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
$$ \begin{align} K_1\ll \left(\frac{R^2N\log (N+M)}{M\log R}\right)^r. \end{align} $$
If
$M=0$
, then we have the bound
$$ \begin{align} K_1\ll \left(\frac{R^2(\log N)^2}{\log R}\right)^r. \end{align} $$
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
$$ \begin{align} m=\sharp {\mathcal I}\leq \left\lfloor t/2\right\rfloor \end{align} $$
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
$$ \begin{align} K_2\ll N^m. \end{align} $$
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
$$ \begin{align} K_3 \le \left(\sharp {\mathcal N}\right)^\ell \ll \left(N\left(N^{-1}+R^{-1}\right)^{1/(d-1)}\right)^{t-m}. \end{align} $$
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
$$ \begin{align} \begin{aligned} \mathsf{M}_s^*(M,N) &\le K_1K_2K_3\\ &\ll \left(R^2(\log N+\log\log N)^2/\log R\right)^r\left(N \log N\right)^m \\ & \qquad\qquad \qquad \left(N\log N\left(N^{-1}+R^{-1}\right)^{1/(d-1)}\right)^{t-m}\\ & \le N^{t+o(1)} R^{2r} \left(\left(N^{-1}+R^{-1}\right)^{1/(d-1)}\right)^{t/2}, \end{aligned} \end{align} $$
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
$$ \begin{align} \prod_{i \in {\mathcal I}}u(n_i)^{k_i} = \prod_{j\in {\mathcal J}} u(n_j)^{k_j},\quad k_i,k_j>0, \end{align} $$
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
$$ \begin{align} \mathrm{rad}(u(n_i))>n_i^{c(\log_2 n_i)/\log_3 n_i}\gg M^{c(\log_2 M)/\log_3(M+N)}. \end{align} $$
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
$$ \begin{align} u(n_j)\equiv 0 \pmod{d_\ell}. \end{align} $$
From (3.8), we have
$$ \begin{align} d_\ell>M^{c_0(\log_2 M)/\log_3(M+N)} \end{align} $$
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
$$ \begin{align} u(m)^a = u(n)^b \end{align} $$
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 [Reference Bugeaud and Kaneko7, 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:
-
(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; -
(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 [Reference Shparlinski and Zannier16], 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
$F\in {\mathbb Z}[X]$
.
Acknowledgements
The authors are very grateful to the referees for the careful reading and valuable comments.
During this work, A.B. and L.H. were supported, in part, by the NKFIH grants 130909 and 150284 and A.O. and I.S. by the Australian Research Council Grant DP230100530. A.O. gratefully acknowledges the hospitality and support of the University of Debrecen, where this work was initiated, and the Max Planck Institute for Mathematics and Institut des Hautes Études Scientifiques, where this work has been carried out.
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