{"paper":{"title":"On the fractional parts of certain sequences of $\\xi \\alpha^{n}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Chi Hoi Yip, Xiang Gao","submitted_at":"2024-08-06T06:06:23Z","abstract_excerpt":"Assume that $\\alpha>1$ is an algebraic number and $\\xi\\neq0$ is a real number. We are concerned with the distribution of the fractional parts of the sequence $(\\xi \\alpha^{n})$. Under various Diophantine conditions on $\\xi$ and $\\alpha$, we obtain lower bounds on the number $n$ with $1\\leq n\\leq N $ for which the fractional part of the sequence $(\\xi \\alpha^{n})_{n\\geq1}$ fall into a prescribed region $I\\subset [0,1]$, extending several results in the literature. As an application, we show that the Fourier decay rate of some self-similar measures is logarithmic, generalizing a result of Varj\\'"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.02972","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/2408.02972/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"}