{"paper":{"title":"Lower bound in the Roth theorem for amenable groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.DS","authors_text":"Pavel Zorin-Kranich, Qing Chu","submitted_at":"2013-09-24T09:31:33Z","abstract_excerpt":"Let $T_1$, $T_2$ be two commuting probability measure-preserving actions of a countable amenable group such that the group spanned by these actions acts ergodically. We show that $\\mu(A\\cap T_1^g A\\cap T_1^g T_2^g A) > \\mu(A)^4-{\\epsilon}$ on a syndetic set for any measurable set $A$ and any $\\epsilon>0$. The proof uses the concept of a sated system introduced by Austin."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6095","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"}