{
  "abstract": "The classical More Sums Than Differences (MSTD) problem studies finite sets $A\\subset\\{0,1,\\ldots,n\\}$ for which $|A+A|>|A-A|$, where $A+A=\\{a_1+a_2:a_1,a_2\\in A\\}$ and $A-A=\\{a_1-a_2:a_1,a_2\\in A\\}$. As addition is commutative and subtraction is not, it was conjectured that as $n\\to\\infty$ almost all subsets $A$ chosen uniformly from the power set of $\\{0,1,\\ldots,n\\}$ are difference-dominated, and it was thus a surprise when Martin and O'Bryant proved a positive percentage of sets are sum-dominant. We greatly generalize this model by introducing a Markov-chain framework, where the classical MSTD model is now just a special case. Let $(X_i)_{i=0}^n$ be a stationary two-state Markov chain on $\\{0,1\\}$ with transition probabilities $P(0,0)=p$ and $P(1,1)=q$, where $p,q\\in(0,1)$. We include $i$ in $A$ exactly when $X_i=1$, and define $A=\\{i\\in\\{0,\\ldots,n\\}:X_i=1\\}$. The usual independent Bernoulli model is recovered when consecutive inclusion decisions are independent, equivalently when $p=1-q$. In particular, the uniformly random subset model corresponds to $p=q=1/2$. Using the fringe-middle method from the MSTD literature, we show that the middle sums and differences are filled with high probability, so the comparison between $|A+A|$ and $|A-A|$ is again governed by endpoint fringes. By fringe manipulation, we prove that the probabilities of sum-dominant, difference-dominant, and balanced sets tend to strictly positive limits as $n\\to\\infty$. We also give numerical estimates of these three probabilities for finite $n$ over a range of values of $p$ and $q$. Through combinatorial methods, we find a closed-form expression for $\\mathbb{E}[|A-A|-|A+A|]$ as $n\\to\\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08394",
  "author_total": 3,
  "authors": [
    {
      "name": "Frank He"
    },
    {
      "name": "Karol Daniewski"
    },
    {
      "name": "Steven J. Miller"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/185hc8sy79vb0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08394",
  "fulltext_url": null,
  "id": "185hc8sy79vb0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08394",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221013371",
  "pdf_url": "https://arxiv.org/pdf/2610.08394",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/185hc8sy79vb0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08394",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08394"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Generating and generalizing MSTD sets through Markov processes",
  "title_text": "Generating and generalizing MSTD sets through Markov processes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/185hc8sy79vb0",
  "venue": "arXiv",
  "versions": []
}