{"paper":{"title":"Linear Diophantine equations in Piatetski-Shapiro sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.NT","authors_text":"Kota Saito, Toshiki Matsusaka","submitted_at":"2020-09-27T17:00:11Z","abstract_excerpt":"A Piatetski-Shapiro sequence with exponent $\\alpha$ is a sequence of integer parts of $n^\\alpha$ $(n = 1,2,\\ldots)$ with a non-integral $\\alpha > 0$. We let $\\mathrm{PS}(\\alpha)$ denote the set of those terms. In this article, we study the set of $\\alpha$ so that the equation $ax + by = cz$ has infinitely many pairwise distinct solutions $(x,y,z) \\in \\mathrm{PS}(\\alpha)^3$, and give a lower bound for its Hausdorff dimension. As a corollary, we find uncountably many $\\alpha > 2$ such that $\\mathrm{PS}(\\alpha)$ contains infinitely many arithmetic progressions of length $3$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.12899","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/2009.12899/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"}