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Jérémy Champagne

Publications and source records attributed to Jérémy Champagne.

4 recordsLinked to original sources
arXiv2026

On the greatest common divisor of $n, \lfloor α_1n\rfloor, \lfloor α_2n^2\rfloor, ..., \lfloor α_kn^k\rfloor$

We answer a question of Bergelson and Richter about the probability of the relation $\gcd(n,\lfloorα_1n\rfloor,\lfloorα_2n^2\rfloor,...,\lfloorα_kn^k\rfloor)=1$ for $n\in\mathbb{N}$ when $α_1,...,α_k$ are fixed irrational numbers. In particular, we avoid the use of exponential sums, as they are difficult to control when one of $α_1,...,α_k$ admits very efficient rational approximations. Instead, we use an elementary method similar to that of Erdős and Lorentz, together with some general bounds on those $n$'s which share a large prime divisor with $\lfloorα_1n\rfloor$.

math.NT↗
arXiv2026

Equidistribution of polynomial sequences in function fields: resolution of a conjecture

Let $\mathbb F_q$ be the finite field of $q$ elements having characteristic $p$, and denote by $\mathbb K_\infty=\mathbb F_q((1/t))$ the field of formal Laurent series in $1/t$. We consider the equidistribution in $\mathbb T=\mathbb K_\infty/\mathbb F_q[t]$ of the values of polynomials $f(u)\in \mathbb K_\infty [u]$ as $u$ varies over $\mathbb F_q[t]$. Let $\mathcal K$ be a finite set of positive integers, and suppose that $α_r\in \mathbb K_\infty$ for $r\in \mathcal K\cup \{0\}$. We show that the polynomial $\sum_{r\in \mathcal K\cup\{0\}}α_ru^r$ is equidistributed in $\mathbb T$ whenever $α_k$ is irrational for some $k\in \mathcal K$ satisfying $p\nmid k$, and also $p^vk\not\in \mathcal K$ for any positive integer $v$. This conclusion resolves in full a conjecture made jointly by the third, fourth and fifth authors.

math.NT↗
arXiv2022

Diophantine approximation with constraints

Following Schmidt, Thurnheer and Bugeaud-Kristensen, we study how Dirichlet's theorem on linear forms needs to be modified when one requires that the vectors of coefficients of the linear forms make a bounded acute angle with respect to a fixed proper non-zero subspace $V$ of $\mathbb{R}^n$. Assuming that the point of $\mathbb{R}^n$ that we are approximating has linearly independent coordinates over $\mathbb{Q}$, we obtain best possible exponents of approximation which surprisingly depend only on the dimension of $V$. Our estimates are derived by reduction to a result of Thurnheer, while their optimality follows from a new general construction in parametric geometry of numbers involving angular constraints.

math.NT↗