{
  "annotation_id": "c080979e-dace-4a32-a876-4617c50c9bbf",
  "date_created": "2026-02-17T05:53:07.897000Z",
  "date_modified": "2026-02-17T05:53:07.897000Z",
  "file_hash": "6149c484e5b3bd11cdc4a49095a15cc4c2d3a61348e414ce0222413a78a189e0",
  "private": false,
  "record": {
    "abstract": "Extending recent developments of Kra, Moreira, Richter and Roberson, we study infinite sumset patterns in $U^k(\\Phi)$-uniform subsets of the integers, defined via the local uniformity seminorms introduced by Host and Kra. The main result relates the degree $k$ of a $U^k(\\Phi)$-uniform set to the existence of sumset patterns along prescribed vertices of $\\ell$-dimensional parallelepipeds, for $k \\leq \\ell$. The proof relies on a dynamical analysis of return-time sets to neighborhoods of points lying over pronilfactor fibers. We then derive higher-order parity obstructions for sumset patterns and consequences in topological dynamics.",
    "arxiv_id": "2601.06915",
    "authors": [
      "Trist\u00e1n Radi\u0107"
    ],
    "categories": [
      "math.DS",
      "math.CO",
      "math.NT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Infinite sumsets in $U^k(\\Phi)$-uniform sets",
    "url": "https://arxiv.org/abs/2601.06915",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "5cf5de80-ece6-40df-9fd5-85c8ab347bf2",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}