{
  "id": "1901.00562",
  "version": "v1",
  "published": "2019-01-03T00:30:11.000Z",
  "updated": "2019-01-03T00:30:11.000Z",
  "title": "Ill-distributed sets over global fields and exceptional sets in Diophantine Geometry",
  "authors": [
    "Marcelo Paredes"
  ],
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $K\\subseteq \\mathbb{R}$ be a number field. Using techniques of discrete analysis, we prove that for definable sets $X$ in $\\mathbb{R}_{\\exp}$ of dimension at most $2$ a conjecture of Wilkie about the density of rational points is equivalent to the fact that $X$ is badly distributed at the level of residue classes for many primes of $K$. This provides a new strategy to prove this conjecture of Wilkie. In order to prove this result, we are lead to study an inverse problem as in the works \\cite{Walsh2, Walsh}, but in the context of number fields, or more generally global fields. Specifically, we prove that if $K$ is a global field, then every subset $S\\subseteq \\mathbb{P}^{n}(K)$ consisting of rational points of projective height bounded by $N$, occupying few residue classes modulo $\\mathfrak{p}$ for many primes $\\mathfrak{p}$ of $K$, must essentially lie in the solution set of a polynomial equation of degree $\\ll (\\log(N))^{C}$, for some constant $C$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-03T00:30:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G50",
      "11G99",
      "11P70",
      "11U09"
    ],
    "keywords": [
      "global field",
      "exceptional sets",
      "diophantine geometry",
      "ill-distributed sets",
      "number field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}