{
  "id": "2004.01884",
  "version": "v1",
  "published": "2020-04-04T07:37:45.000Z",
  "updated": "2020-04-04T07:37:45.000Z",
  "title": "$L$--functions and sum--free sets",
  "authors": [
    "Tomasz Schoen",
    "Ilya D. Shkredov"
  ],
  "comment": "15 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "For set $A\\subset {\\mathbb {F}_p}^*$ define by ${\\mathsf{sf}}(A)$ the size of the largest sum--free subset of $A.$ Alon and Kleitman showed that ${\\mathsf{sf}} (A) \\ge |A|/3+O(|A|/p).$ We prove that if ${\\mathsf{sf}} (A)-|A|/3$ is small then the set $A$ must be uniformly distributed on cosets of each large multiplicative subgroup. Our argument relies on irregularity of distribution of multiplicative subgroups on certain intervals in ${\\mathbb {F}_p}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-04T07:37:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "11B50",
      "11B75",
      "11M06"
    ],
    "keywords": [
      "sum-free sets",
      "largest sum-free subset",
      "argument relies",
      "large multiplicative subgroup",
      "irregularity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}