{
  "id": "1906.03793",
  "version": "v1",
  "published": "2019-06-10T04:56:50.000Z",
  "updated": "2019-06-10T04:56:50.000Z",
  "title": "Union of Two Arithmetic Progressions with the Same Common Difference Is Not Sum-dominant",
  "authors": [
    "Hung Viet Chu"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Given a finite set $A\\subseteq \\mathbb{N}$, define the sum set $$A+A = \\{a_i+a_j\\mid a_i,a_j\\in A\\}$$ and the difference set $$A-A = \\{a_i-a_j\\mid a_i,a_j\\in A\\}.$$ The set $A$ is said to be sum-dominant if $|A+A|>|A-A|$. We prove the following results 1) The union of two arithmetic progressions (with the same common difference) is not sum-dominant. This result partially proves a conjecture proposed by the author in a previous paper; that is, the union of any two arbitrary arithmetic progressions is not sum-dominant. The author is motivated by the anonymous referee's comment that the conjecture was marvelous and tantalizing. 2) Hegarty proved that a sum-dominant set must have at least $8$ elements with computers' help. The author of the current paper provided a human-verifiable proof that a sum-dominant set must have at least $7$ elements. A natural question is about the largest cardinality of sum-dominant subsets of an arithmetic progression. Fix $n\\ge 16$. Let $N$ be the cardinality of the largest sum-dominant subset(s) of $\\{0,1,\\ldots,n-1\\}$ that contain(s) $0$ and $n-1$. Then $n-7\\le N\\le n-4$; that is, from an arithmetic progression of length $n\\ge 16$, we need to discard at least $4$ and at most $7$ elements (in a clever way) to have the largest sum-dominant set(s). 3) Let $R\\in \\mathbb{N}$ have the property that for all $r\\ge R$, $\\{1,2,\\ldots,r\\}$ can be partitioned into $3$ sum-dominant subsets, while $\\{1,2,\\ldots,R-1\\}$ cannot. Then $24\\le R\\le 145$. This result answers a question by the author et al. in another paper on whether we can find a stricter upper bound for $R$. The previous upper bound was at least $888$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-10T04:56:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P99"
    ],
    "keywords": [
      "common difference",
      "largest sum-dominant subset",
      "stricter upper bound",
      "largest sum-dominant set",
      "arbitrary arithmetic progressions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}