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Jean-Marc Deshouillers
Quand seule la sous-somme vide est nulle modulo ${p}$
Journal de théorie des nombres de Bordeaux, 19 no. 1 (2007), p. 71-79, doi: 10.5802/jtnb.574
Article PDF | Reviews MR 2332054 | Zbl 1153.11007

Résumé - Abstract

Let $c>1$, $p$ be a prime number and $\mathcal{A}$ a subset of $\mathbb{Z}/p\mathbb{Z}$ with cardinality larger than $c\sqrt{p}$ and such that for any non empty subset $\mathcal{B}$ of $\mathcal{A}$, one has $\sum _{b \in \mathcal{B}} b \ne 0$. We show that there exists $s$ coprime with $p$ such that the set $s.\mathcal{A}$ is very concentrated around the origin, and that it is almost exclusively composed of elements with a positive fractional part. More precisely, one has

$$\sum _{a \in \mathcal{A}} \left\Vert \frac{sa}{p} \right\Vert < 1 + O(p^{-1/4} \ln p) \;\;\;\text{and}\!\! \sum _{\begin{array}{c}a \in \mathcal{A},\\ \lbrace sa/p\rbrace \ge 1/2\end{array}} \left\Vert \frac{sa}{p} \right\Vert = O(p^{-1/4} \ln p).$$

We also show that the error terms cannot be replaced by $o(p^{-1/2})$.

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