{
  "abstract": "By introducing new ergodic-theoretic techniques in nilsystems, we determine which infinite sumset configurations occur in $U^k(Φ)$-uniform and Nil-Bohr sets. To be more precise, our first result associates the degree $k$ of a $U^k(Φ)$-uniform set with the variety of sumsets it contains, solving a conjecture of Kra, Moreira, Richter and Robertson. Restricting to Nil-Bohr sets we show the existence of infinite sumsets with summands in the shifted primes $\\mathbb{P}-1$. As a consequence, we show that for any real polynomial $Q(n)$ with leading irrational coefficient of degree $k$, and any natural numbers $\\ell_1, \\cdots, \\ell_k$ there is an infinite set $P\\subset \\mathbb{P}$ such that \\begin{equation*} Q\\Big(\\sum_{p \\in I} p\\Big) \\in U \\pmod 1 \\quad \\text{ for all } I \\subset P , |I| = \\ell_1, \\ldots, \\ell_k. \\end{equation*}",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20522",
  "author_total": 2,
  "authors": [
    {
      "name": "Felipe Hernández"
    },
    {
      "name": "Tristán Radić"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3wfnksqge10vv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20522",
  "fulltext_url": null,
  "id": "3wfnksqge10vv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20522",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7213612349",
  "pdf_url": "https://arxiv.org/pdf/2609.20522",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3wfnksqge10vv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20522",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20522"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Infinite prime sumsets in structured and $U^k(Φ)$-uniform sets",
  "title_text": "Infinite prime sumsets in structured and U^k(Φ)-uniform sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/3wfnksqge10vv",
  "venue": "arXiv",
  "versions": []
}