{"paper":{"title":"The entropy function of an invariant measure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.PR"],"primary_cat":"math.LO","authors_text":"Cameron Freer, Nathanael Ackerman, Rehana Patel","submitted_at":"2018-09-07T03:07:31Z","abstract_excerpt":"Given a countable relational language $L$, we consider probability measures on the space of $L$-structures with underlying set $\\mathbb{N}$ that are invariant under the logic action. We study the growth rate of the entropy function of such a measure, defined to be the function sending $n \\in \\mathbb{N}$ to the entropy of the measure induced by restrictions to $L$-structures on $\\{0, \\ldots, n-1\\}$. When $L$ has finitely many relation symbols, all of arity $k\\ge 1$, and the measure has a property called non-redundance, we show that the entropy function is of the form $Cn^k+o(n^k)$, generalizing"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.02290","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"}