{
  "id": "2011.09306",
  "version": "v1",
  "published": "2020-11-16T23:11:01.000Z",
  "updated": "2020-11-16T23:11:01.000Z",
  "title": "Metric theory of Weyl sums",
  "authors": [
    "Changhao Chen",
    "Bryce Kerr",
    "James Maynard",
    "Igor Shparlinski"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2004.02539",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove that there exist positive constants $C$ and $c$ such that for any integer $d \\ge 2$ the set of ${\\mathbf x}\\in [0,1)^d$ satisfying $$ cN^{1/2}\\le \\left|\\sum^N_{n=1}\\exp\\left (2 \\pi i \\left (x_1n+\\ldots+x_d n^d\\right)\\right) \\right|\\le C N^{1/2}$$ for infinitely many natural numbers $N$ is of full Lebesque measure. This substantially improves the previous results where similar sets have been measured in terms of the Hausdorff dimension. We also obtain similar bounds for exponential sums with monomials $xn^d$ when $d\\neq 4$. Finally, we obtain lower bounds for the Hausdorff dimension of large values of general exponential polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-16T23:11:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weyl sums",
      "metric theory",
      "hausdorff dimension",
      "general exponential polynomials",
      "full lebesque measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}