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The f-rotation set is Gamma_f={\\alpha \\in G: M_N^{\\alpha} f(x) converges for m a.e. x as N\\to \\infty .} We prove that if G is a compact locally connected Abelian group and f: G -> R is a measurable function then from m(Gamma_f)>0 it follows that f \\in L^1(G). A similar result is established for ordinary Birkhoff averages if G=Z_{p}, the group "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1411.0508","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-11-03T14:44:42Z","cross_cats_sorted":["math.CA","math.GR"],"title_canon_sha256":"c364e1352a9aed58ab6c59258e3688077b4f924d3919da5d94323a06ff81dd9f","abstract_canon_sha256":"1a57815ced3bf49a257f0dcc7784635cf2ea9bfd5e20dff030fd0fc086e54941"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:14.089044Z","signature_b64":"MRIrcGyQ2UipS5HIpCg8xctrmjxU4j+fMx8JMIprD6BnXZ1CDvJHV7l/PCJBZ3JUPq5UQZPdKBJc3vU3wm0jBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"57866970106b56ee6fcde5afd03040383d6ce14563fba9ac16a695bb1e120a60","last_reissued_at":"2026-05-17T23:53:14.088274Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:14.088274Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Convergence of ergodic averages for many group rotations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.GR"],"primary_cat":"math.DS","authors_text":"Gabriella Keszthelyi, Zoltan Buczolich","submitted_at":"2014-11-03T14:44:42Z","abstract_excerpt":"Suppose that G is a compact Abelian topological group, m is the Haar measure on G and f is a measurable function. Given (n_k), a strictly monotone increasing sequence of integers we consider the nonconventional ergodic/Birkhoff averages M_N^{\\alpha}f(x). The f-rotation set is Gamma_f={\\alpha \\in G: M_N^{\\alpha} f(x) converges for m a.e. x as N\\to \\infty .} We prove that if G is a compact locally connected Abelian group and f: G -> R is a measurable function then from m(Gamma_f)>0 it follows that f \\in L^1(G). 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