{
  "id": "1510.05411",
  "version": "v1",
  "published": "2015-10-19T10:05:40.000Z",
  "updated": "2015-10-19T10:05:40.000Z",
  "title": "Discrete spheres and arithmetic progressions in product sets",
  "authors": [
    "Dmitrii Zhelezov"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove that if $B$ is a set of $N$ positive integers such that $B\\cdot B$ contains an arithmetic progression of length $M$ then $N\\geq \\pi(M) + M^{2/3-o(1)}$. On the other hand, there are examples for which $N< \\pi(M)+ M^{2/3}$. This improves previously known bounds of the form $N = \\Omega(\\pi(M))$ and $N = O(\\pi(M))$, respectively. The main new tool is a reduction of the original problem to the question of an approximate additive decomposition of the $3$-sphere in $\\mathbb{F}_3^n$ which is the set of 0-1 vectors with exactly three non-zero coordinates. Namely, we prove that such a set cannot be contained in a sumset $A+A$ unless $|A| \\gg n^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-19T10:05:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B25"
    ],
    "keywords": [
      "arithmetic progression",
      "product sets",
      "discrete spheres",
      "approximate additive decomposition",
      "original problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151005411Z"
    }
  }
}