{
  "id": "2401.10152",
  "version": "v2",
  "published": "2024-01-18T17:27:34.000Z",
  "updated": "2024-01-20T23:00:38.000Z",
  "title": "Sums of square roots that are close to an integer",
  "authors": [
    "Stefan Steinerberger"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $k \\in \\mathbb{N}$ and suppose we are given $k$ integers $1 \\leq a_1, \\dots, a_k \\leq n$. If $\\sqrt{a_1} + \\dots + \\sqrt{a_k}$ is not an integer, how close can it be to one? When $k=1$, the distance to the nearest integer is $\\gtrsim n^{-1/2}$. Angluin-Eisenstat observed the bound $\\gtrsim n^{-3/2}$ when $k=2$. We prove there is a universal $c>0$ such that, for all $k \\geq 2$, there exists a $c_k > 0$ and $k$ integers in $\\left\\{1,2,\\dots, n\\right\\}$ with $$ 0 <\\|\\sqrt{a_1} + \\dots + \\sqrt{a_k} \\| \\leq c_k\\cdot n^{-c \\cdot k^{1/3}},$$ where $\\| \\cdot \\|$ denotes the distance to the nearest integer. This is a case of the square-root sum problem in numerical analysis where the usual cancellation constructions do not apply: even for $k=3$, constructing explicit examples of integers whose square root sum is nearly an integer appears to be nontrivial.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-01-20T23:00:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nearest integer",
      "square-root sum problem",
      "usual cancellation constructions",
      "square root sum",
      "constructing explicit examples"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}