{
  "id": "2005.13432",
  "version": "v1",
  "published": "2020-05-27T15:42:30.000Z",
  "updated": "2020-05-27T15:42:30.000Z",
  "title": "Sum-product estimates for diagonal matrices",
  "authors": [
    "Akshat Mudgal"
  ],
  "comment": "A revised version will appear in Bulletin of the Australian Mathematical Society. 9 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Given $d \\in \\mathbb{N}$, we establish sum-product estimates for finite, non-empty subsets of $\\mathbb{R}^d$. This is equivalent to a sum-product result for sets of diagonal matrices. In particular, let $A$ be a finite, non-empty set of $d \\times d$ diagonal matrices with real entries. Then for all $\\delta_1 < 1/3 + 5/5277$, we have \\[ |A+A| + |A\\cdot A| \\gg_{d} |A|^{1 + \\delta_{1}/d}. \\] In this setting, the above estimate quantitatively strengthens a result of Chang.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-27T15:42:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30"
    ],
    "keywords": [
      "diagonal matrices",
      "non-empty subsets",
      "sum-product result",
      "real entries",
      "estimate quantitatively strengthens"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}