{"paper":{"title":"Fourier coefficients of $\\times p$-invariant measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Huichi Huang","submitted_at":"2016-06-20T12:09:49Z","abstract_excerpt":"We consider densities $D_\\Sigma(A)$, $\\overline{D}_\\Sigma(A)$ and $\\underline{D}_\\Sigma(A)$ for a subset $A$ of $\\mathbb{N}$ with respect to a sequence $\\Sigma$ of finite subsets of $\\mathbb{N}$ and study Fourier coefficients of ergodic, weakly mixing and strongly mixing $\\times p$-invariant measures on the unit circle $\\mathbb{T}$. Combining these, we prove the following measure rigidity results: on $\\mathbb{T}$, the Lebesgue measure is the only non-atomic $\\times p$-invariant measure satisfying one of the following: (1) $\\mu$ is ergodic and there exist a F\\o lner sequence $\\Sigma$ in $\\mathb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.06078","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"}