{
  "id": "1808.05460",
  "version": "v1",
  "published": "2018-08-16T13:27:06.000Z",
  "updated": "2018-08-16T13:27:06.000Z",
  "title": "Infinite Families of Partitions into MSTD Subsets",
  "authors": [
    "Hung Viet Chu",
    "Noah Luntzlara",
    "Steven J. Miller",
    "Lily Shao"
  ],
  "comment": "Version 1.0, 18 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "A set $A$ is MSTD (more-sum-than-difference) if $|A+A|>|A-A|$. Though MSTD sets are rare, Martin and O'Bryant proved that there exists a positive constant lower bound for the proportion of MSTD subsets of $\\{1,2,\\ldots ,r\\}$ as $r\\rightarrow\\infty$. Asada et al. [AMMS] showed that there exists a positive constant lower bound for the proportion of decompositions of $\\{1,2,\\ldots,r\\}$ into two MSTD subsets as $r\\rightarrow\\infty$, which implies the result of Martin and O'Bryant. However, the method is probabilistic and does not give explicit decompositions. Continuing this work, we provide an efficient method to partition $\\{1,2,\\ldots,r\\}$ (for $r$ sufficiently large) into $k \\ge 2$ MSTD subsets, positively answering a question raised in [AMMS] as to whether or not this is possible for all such $k$. Next, let $R$ be the smallest integer such that for all $r\\ge R$, $\\{1,2,\\ldots,r\\}$ can be $k$-decomposed into MSTD subsets, while $\\{1,2,\\ldots,R-1\\}$ cannot be $k$-decomposed into MSTD subsets. We establish rough lower and upper bounds for $R$ and the gap between the two bounds grows linearly with $k$. Lastly, we provide a sufficient condition on when there exists a constant lower bound for the proportion of decompositions of $\\{1,2,\\ldots,r\\}$ into $k$ MSTD subsets as $r\\rightarrow \\infty$. This condition offers an alternative proof of Theorem 1.4 in [AMMS] and can be a promising approach to generalize the",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-16T13:27:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P99",
      "11K99"
    ],
    "keywords": [
      "mstd subsets",
      "infinite families",
      "positive constant lower bound",
      "proportion",
      "explicit decompositions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}