{"paper":{"title":"Non-Salem sets in multiplicative Diophantine approximation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bo Tan, Qing-Long Zhou","submitted_at":"2024-09-19T08:25:43Z","abstract_excerpt":"In this paper, we answer a question of Cai-Hambrook in (arXiv$\\colon$ 2403.19410). Furthermore, we compute the Fourier dimension of the multiplicative $\\psi$-well approximable set $$M_2^{\\times}(\\psi)=\\left\\{(x_1,x_2)\\in [0,1]^{2}\\colon \\|qx_1\\|\\|qx_2\\|<\\psi(q) \\text{ for infinitely many } q\\in \\N\\right\\},$$ where $\\psi\\colon\\N\\to [0,\\frac{1}{4})$ is a positive function satisfying $\\sum_q\\psi(q)\\log\\frac{1}{\\psi(q)}<\\infty.$ As a corollary, we show that the set $M_2^{\\times}(q\\mapsto q^{-\\tau})$ is non-Salem for $\\tau>1.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.12557","kind":"arxiv","version":1},"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/2409.12557/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"}