{"paper":{"title":"Distributions of Finite Sequences Represented by Polynomials in Piatetski-Shapiro Sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kota Saito, Yuuya Yoshida","submitted_at":"2020-06-24T17:58:48Z","abstract_excerpt":"By using the work of Frantzikinakis and Wierdl, we can see that for all $d\\in\\mathbb{N}$, $\\alpha\\in(d,d+1)$, and integers $k\\ge d+2$ and $r\\ge1$, there exist infinitely many $n\\in\\mathbb{N}$ such that the sequence $(\\lfloor{(n+rj)^\\alpha}\\rfloor)_{j=0}^{k-1}$ is represented as $\\lfloor{(n+rj)^\\alpha}\\rfloor=p(j)$, $j=0,1,\\ldots,k-1$, by using some polynomial $p(x)\\in\\mathbb{Q}[x]$ of degree at most $d$. In particular, the above sequence is an arithmetic progression when $d=1$. In this paper, we show the asymptotic density of such numbers $n$ as above. When $d=1$, the asymptotic density is equ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.13930","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2006.13930/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}