{"docId":85,"paperId":85,"url":"https:\/\/hrj.episciences.org\/85","doi":"10.46298\/hrj.2005.85","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[{"vid":39,"name":"Volume 28 - 2005"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-01110947","repositoryVersion":1,"repositoryLink":"https:\/\/hal.archives-ouvertes.fr\/hal-01110947v1","dateSubmitted":"2015-03-03 16:13:33","dateAccepted":"2015-06-12 16:05:09","datePublished":"2005-01-01 08:00:00","titles":{"en":"A lower bound concerning subset sums which do not cover all the residues modulo $p$."},"authors":["Deshouillers, Jean-Marc"],"abstracts":{"en":"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 $$ \\sum_{a\\in\\mathcal A}\\Vert\\frac{as}{p}\\Vert>1+Kp^{-1\/2}.$$"},"keywords":[{"en":"upper bound for the error term"},{"en":"residue classes modulo $p$"},"[MATH]Mathematics [math]"]}