{
  "id": "1902.05757",
  "version": "v1",
  "published": "2019-02-15T10:36:33.000Z",
  "updated": "2019-02-15T10:36:33.000Z",
  "title": "On semilinear sets and asymptotically approximate groups",
  "authors": [
    "Arindam Biswas",
    "Wolfgang Alexander Moens"
  ],
  "categories": [
    "math.NT",
    "math.CO",
    "math.GR"
  ],
  "abstract": "Let $G$ be any group and $A$ be an arbitrary subset of $G$ (not necessarily symmetric and not necessarily containing the identity). The $h$-fold product set of $A$ is defined as $$A^{h} :=\\lbrace a_{1}.a_{2}...a_{h} : a_{1},\\ldots,a_n \\in A \\rbrace.$$ Nathanson considered the concept of an asymptotic approximate group. Let $r,l \\in \\mathbb{N}$. The set $A$ is said to be an $(r,l)$ approximate group in $G$ if there exists a subset $X$ in $G$ such that $|X|\\leqslant l$ and $A^{r}\\subseteq XA$. The set $A$ is an asymptotic $(r,l)$-approximate group if the product set $A^{h}$ is an $(r,l)$-approximate group for all sufficiently large $h$. Recently, Nathanson showed that every finite subset $A$ of an abelian group is an asymptotic $(r,l')$ approximate group (with the constant $l'$ explicitly depending on $r$ and $A$). We generalise the result and show that, in an arbitrary abelian group $G$, the union of $k$ (unbounded) generalised arithmetic progressions is an asymptotic $(r,(4rk)^k)$-approximate group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-15T10:36:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "05A18",
      "11B75",
      "11P70",
      "20K99"
    ],
    "keywords": [
      "asymptotically approximate groups",
      "semilinear sets",
      "fold product set",
      "asymptotic approximate group",
      "arbitrary abelian group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}