episciences.org_85_1653159372
1653159372
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
HardyRamanujan Journal
28047370
10.46298/journals/hrj
https://hrj.episciences.org
01
01
2005
Volume 28  2005
A lower bound concerning subset sums which do not cover all the residues modulo $p$.
JeanMarc
Deshouillers
Let $c>\sqrt{2}$ and let $p$ be a prime number. JM. 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 $$ \sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert>1+Kp^{1/2}.$$
01
01
2005
85
https://hal.archivesouvertes.fr/hal01110947v1
10.46298/hrj.2005.85
https://hrj.episciences.org/85

https://hrj.episciences.org/85/pdf