{"paper":{"title":"Finiteness of solutions to linear Diophantine equations on Piatetski-Shapiro sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kota Saito","submitted_at":"2023-06-30T17:23:02Z","abstract_excerpt":"A sequence of integers of the form $\\lfloor n^{\\alpha}\\rfloor$ $(n=1,2,\\ldots)$ for some fixed non-integral $\\alpha>1$ is called a Piatetski-Shapiro sequence, where $\\lfloor x\\rfloor$ denotes the integer part of $x$. Let $\\mathrm{PS}(\\alpha)$ denote the set of all those terms. In this article, we show that $x+y=z$ has only finitely many solutions $(x,y,z)\\in \\mathrm{PS}(\\alpha)^3$ for almost every $\\alpha>3$. Furthermore, we show that $\\mathrm{PS}(\\alpha)$ has only finitely many arithmetic progressions of length $3$ for almost every $\\alpha>10$. In addition, we estimate upper bounds for the Ha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.17813","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/2306.17813/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"}