{
  "abstract": "We establish the first pointwise convergence result for nonconventional ergodic averages modeled along a large class of oscillating sequences, treating, for example, orbits of the form $\\big(\\lfloor n^d\\sin(eπn)\\rfloor\\big)_{n\\in\\mathbb{N}}$, $d\\in\\mathbb{N}$. More specifically, for every probability space $(X,\\mathcal{B},μ)$ equipped with an invertible measure-preserving transformation $T\\colon X\\to X$ and every $f\\in L^p_μ(X)$ with $p\\in(1,\\infty]$, we prove that the limit \\[ \\lim_{N\\to\\infty}\\frac{1}{N}\\sum_{n=1}^Nf\\big(T^{\\lfloor P(n)ψ(αn)\\rfloor}x\\big)\\quad\\text{exists for $μ$-a.e. $x\\in X$,} \\] where $P$ is a non-constant polynomial with real coefficients, $ψ$ is $1$-periodic and real-analytic, and $α$ has irrationality exponent smaller than $1+\\sqrt{\\operatorname{deg}(P)+1}$. On the one hand, the latter assumption is satisfied by a.e. $α\\in\\mathbb{R}$, and, on the other hand, our result covers any of the following concrete orbits \\[ \\lfloor n\\cos(eπn)\\rfloor\\text{,}\\quad\\lfloor n^{37}\\sin(n)\\rfloor\\text{,}\\quad\\big\\lfloor \\sqrt{2}n^{10}e^{\\sin(π(\\sqrt[5]{7}+1) n)}\\big\\rfloor\\text{,}\\quad \\big\\lfloor \\big((\\log 2) n^3+e n\\big)e^{e^{\\cos(π\\sqrt{2} n)}}\\big\\rfloor\\text{.} \\] The proof employs the so-called one-frequency circle method for estimating exponential sums with phases $\\big(ξ\\lfloor P(n)ψ(αn)\\rfloor\\big)_{1\\le n\\le N}$, but the major arc approximants of our exponential sums do not consist of their direct continuous counterparts. In the major arc regime we decouple the $ψ$-oscillatory part of the phase from the polynomial one, and approximate the exponential sum by $\\int_0^1\\big(\\frac{1}{N}\\int_0^Ne^{2πiξP(t)ψ( s)}dt\\big)ds$. This represents a novel phenomenon for problems of this kind, and, interestingly, our proof exploits the coefficient-free variational estimates of Radon-type operators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34301",
  "author_total": 1,
  "authors": [
    {
      "name": "Leonidas Daskalakis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/94gp9bwz3k2f8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34301",
  "fulltext_url": null,
  "id": "94gp9bwz3k2f8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34301",
  "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.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7223883414",
  "pdf_url": "https://arxiv.org/pdf/2610.34301",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/94gp9bwz3k2f8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34301",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34301"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Pointwise ergodic theorems along oscillating sequences",
  "title_text": "Pointwise ergodic theorems along oscillating sequences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/94gp9bwz3k2f8",
  "venue": "arXiv",
  "versions": []
}