10.46298/hrj.2005.152
https://hrj.episciences.org/152
Deshouillers, Jean-Marc
Jean-Marc
Deshouillers
A lower bound concerning subset sums which do not cover all the residues modulo $p$.
Let $c>\sqrt{2}$ and let $p$ be a prime number. J-M. Deshouillers and G. A. Freiman proved that a subset $\mathcal A$ of $\mathbb{Z}/p\mathbb{Z}$, with cardinality larger than $c\sqrt{p}$ and such that its subset sums do not cover $\mathbb{Z}/p\mathbb{Z}$ has an isomorphic image which is rather concentrated; more precisely, there exists $s$ prime to $p$ such that $$\sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert < 1+O(p^{-1/4}\ln p),$$ where the constant implied in the ``O'' symbol depends on $c$ at most. We show here that there exist a $K$ depending on $c$ at most, and such sets $\mathcal A$, such that for all $s$ prime to $p$ one has $$ \sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert>1+Kp^{-1/2}.$$
episciences.org
upper bound for the error term
residue classes modulo $p$
[MATH] Mathematics [math]
2015-06-12
2005-01-01
2005-01-01
en
journal article
https://hal.archives-ouvertes.fr/hal-01110947v1
2804-7370
https://hrj.episciences.org/152/pdf
VoR
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Hardy-Ramanujan Journal
Volume 28 - 2005
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