arXiv subjects

Jean-Marc Deshouillers

Publications and source records attributed to Jean-Marc Deshouillers.

14 recordsLinked to original sources

arXiv2025

On coprimality of consecutive elements in certain sequences

The study of finding blocks of primes in certain arithmetic sequences is one of the classical problems in number theory. It is also very interesting to study blocks of consecutive elements in such sequences that are pairwise coprime. In this context, we show that if $f$ is a twice continuously differentiable real-valued function on $[1, \infty)$ such that $f''(x) \to 0$ as $x \to \infty$ and $\limsup_{x \to \infty} f'(x) = \infty$, then there exist arbitrarily long blocks of pairwise coprime consecutive elements in the sequence $(\lfloor f(n) \rfloor)_n$. This result refines the qualitative part of a recent result by the first author, Drmota and Müllner. We also prove that there exists a subset $\mathcal{A} \subseteq \mathbb{N}$ having upper Banach density one such that for any two distinct integers $m, n \in \mathcal{A}$, the integers $\lfloor f(m) \rfloor$ and $\lfloor f(n) \rfloor$ are pairwise coprime. Further, we show that there exist arbitrarily long blocks of consecutive elements in the sequence $(\lfloor f(n) \rfloor)_n$ such that no two of them are pairwise coprime.

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arXiv2025

Coprimality of elements in regular sequences with polynomial growth

The investigation of primes in certain arithmetic sequences is one of the fundamental problems in number theory and especially, finding blocks of distinct primes has gained a lot of attention in recent years. In this context, we prove the existence of long blocks of $k$-wise coprime elements in certain regular sequences. More precisely, we prove that for any positive integers $H \geq k \geq 2$ and for a real-valued $k$-times continuously differentiable function $f \in \mathcal{C}^k\left( [1, \infty)\right)$ satisfying $\lim_{x \to \infty} f^{(k)}(x) = 0$ and $\limsup_{x \to \infty} f^{(k-1)}(x) = \infty$, there exist infinitely many positive integers $n$ such that $$ \gcd\left( \lfloor f(n+i_1)\rfloor, \lfloor f(n+i_2)\rfloor, \cdots, \lfloor f(n+i_k)\rfloor \right) ~=~ 1 $$ for any integers $1 \leq i_1 < i_2 < \cdots < i_k \leq H$. Further, we show that there exists a subset $\mathcal{A} \subseteq \mathbb{N}$ having upper Banach density one such that $$ \gcd\left(\lfloor f(n_1) \rfloor, \lfloor f(n_2) \rfloor, \cdots, \lfloor f(n_k) \rfloor\right) ~=~ 1 $$ for any distinct integers $n_1, n_2, \cdots, n_k \in \mathcal{A}$.

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arXiv2024

Binary-ternary collisions and the last significant digit of $n!$ in base 12

The third-named author recently proved [Israel J. of Math. 258 (2023), 475--502] that there are infinitely many \textit{collisions} of the base-2 and base-3 sum-of-digits functions. In other words, the equation \[ s_2(n)=s_3(n) \] admits infinitely many solutions in natural numbers. We refine this result and prove that every integer $a$ in $\{1, 2, \ldots, 11\}$ appears as the last nonzero digit of $n!$ in base $12$ infinitely often.

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arXiv2022

Synchronizing automatic sequences along Piatetski-Shapiro sequences

The purpose of this paper is to study subsequences of synchronizing $k$-automatic sequences $a(n)$ along Piatetski-Shapiro sequences $\lfloor n^c \rfloor$ with non-integer $c>1$. In particular, we show that $a(\lfloor n^c \rfloor)$ satisfies a prime number theorem of the form $\sum_{n\le x} Λ(n)a(\lfloor n^c \rfloor) \sim C\, x$, and, furthermore, that it is deterministic for $c \in \mathbb R\setminus \mathbb Z$. As an interesting additional result, we show that the sequence $\lfloor n^c\rfloor \bmod m$ has polynomial subword complexity.

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arXiv2021

On a question of Luca and Schinzel over Segal-Piatetski-Shapiro sequences

We extend to Segal-Piatetski-Shapiro sequences previous results on the Luca-Schinzel question over integral valued polynomial sequences. Namely, we prove that for any real $c$ larger than $1$ the sequence $(\sum_{m\le n} φ(\lfloor m^c \rfloor) /\lfloor m^c \rfloor)_n$ is dense modulo $1$, where $φ$ denotes Euler's totient function. The main part of the proof consists in showing that when $R$ is a large integer, the sequence of the residues of $\lfloor m^c \rfloor$ modulo $R$ contains blocks of consecutive values which are in an arithmetic progression.

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arXiv2021

On the local structure of the set of values of Euler's $φ$ function

Assuming the validity of Dickson's conjecture, we show that the set $\mathcal{V}$ of values of the Euler's totient function $φ$ contains arbitrarily large arithmetic progressions with common difference 4. This leads to the question of proving unconditionally that this set $\mathcal{V}$ has a positive upper Banach density.

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arXiv2018

Randomness and non-randomness properties of Piatetski-Shapiro sequences modulo m

We study Piatetski-Shapiro sequences $(\lfloor n^c\rfloor)_n$ modulo m, for non-integer $c >1$ and positive $m$, and we are particularly interested in subword occurrences in those sequences. We prove that each block $\in\{0,1\}^k$ of length $k < c + 1$ occurs as a subword with the frequency $2^{-k}$, while there are always blocks that do not occur. In particular, those sequences are not normal. For $1<c<2$, we estimate the number of subwords from above and below, yielding the fact that our sequences are deterministic and not morphic. Finally, using the Daboussi-Kátai criterion, we prove that the sequence $\lfloor n^c\rfloor$ modulo m is asymptotically orthogonal to multiplicative functions bounded by $1$ and with mean value $0$.

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arXiv2018

Integral points on convex curves

We estimate the maximal number of integral points which can be on a convex arc in the plane with given length, minimal radius of curvature and initial slope.

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arXiv2018

Random ordering in modulus of consecutive Hecke eigenvalues of primitive forms

Let τ(.) be the Ramanujan τ-function, and let k be a positive integer such that τ(n) is not 0 for n=1,...,[k/2]. (This is known to be true for k < 10^{23}, and, conjecturally, for all k.) Further, let s be a permutation of the set {1,...,k}. Then there exist infinitely many positive integers m such that |τ(m+s(1))|<τ(m+s(2))|<...<|τ(m+s(k))|. We also obtain a similar result for Fourier-coefficients of general newforms.

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arXiv2016

Sums of the digits in bases 2 and 3

Let b $\ge$ 2 be an integer and let s b (n) denote the sum of the digits of the representation of an integer n in base b. For sufficiently large N , one has Card{n $\le$ N : |s 3 (n) -- s 2 (n)| $\le$ 0.1457205 log n} \textgreater{} N 0.970359. The proof only uses the separate (or marginal) distributions of the values of s 2 (n) and s 3 (n).

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arXiv2014

Additive properties of sequences of pseudo s-th powers

In this paper, we study (random) sequences of pseudo s-th powers, as introduced by Erdös and Rényi in 1960. In 1975, Goguel proved that such a sequence is almost surely not an asymptotic basis of order s. Our first result asserts that it is however almost surely a basis of order s + x for any x > 0. We then study the s-fold sumset sA = A + ... + A (s times) and in particular the minimal size of an additive complement, that is a set B such that sA + B contains all large enough integers. With respect to this problem, we prove quite precise theorems which are tantamount to asserting that a threshold phenomenon occurs.

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arXiv2014

Gaps in sumsets of $s$ pseudo s-th power sequences

We study the length of the gaps between consecutive members in the sumset sA when A is a pseudo s-th power sequence, with s>1. We show that, almost surely, limsup (b_{n+1}-b_{n})/log (b_n) = s^s s!/Γ^s(1/s), where b_n are the elements of sA.

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arXiv2009

Large zero-free subsets of Z/pZ

A finite subset $A$ of an abelian group $G$ is said to be zero-free if the identity element of $G$ cannot be written as a sum of distinct elements from $A$. In this article we study the structure of zero-free subsets of $Z/pZ$ the cardinality of which is close to largest possible. In particular, we determine the cardinality of the largest zero-free subset of $Z/pZ$, when $p$ is a sufficiently large prime.

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arXiv2007

Refined bound for sum-free sets in groups of prime order

Improving upon earlier results of Freiman and the present authors, we show that if $p$ is a sufficiently large prime and $A$ is a sum-free subset of the group of order $p$, such that $n:=|A|>0.318p$, then $A$ is contained in a dilation of the interval $[n,p-n]\pmod p$.

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