{"paper":{"title":"Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.GR"],"primary_cat":"math.DS","authors_text":"Catalin Badea, Sophie Grivaux","submitted_at":"2018-04-04T12:40:16Z","abstract_excerpt":"Exploiting a construction of rigidity sequences for weakly mixing dynamical systems by Fayad and Thouvenot, we show that for every integers $p_{1},\\dots,p_{r}$ there exists a continuous probability measure $\\mu $ on the unit circle $\\mathbb{T}$ such that \\[ \\inf_{k_{1}\\ge 0,\\dots,k_{r}\\ge 0}|\\widehat{\\mu }(p_{1}^{k_{1}}\\dots p_{r}^{k_{r}})|>0. \\] This results applies in particular to the Furstenberg set $F=\\{2^{k}3^{k'}\\,;\\,k\\ge 0,\\ k'\\ge 0\\}$, and disproves a 1988 conjecture of Lyons inspired by Furstenberg's famous $\\times 2$-$\\times 3$ conjecture. We also estimate the modified Kazhdan const"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.01369","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":""},"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"}