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In this paper we will show that $| \\sup_{j} |S_{n_j}(f)| |_{1,\\infty}\\leq C |f|_{1} \\log\\log (10+\\frac{|f|_{\\infty}}{|f|_1})\\:.$ As a direct consequence we obtain that $S_{n_j}f \\rightarrow f $ a.e. for $f\\in L\\log\\log L\\log\\log\\log L$. The (discrete) Walsh model version of this last fact was proved by Do and Lacey but their methods do no"},"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":"1204.6542","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-04-30T03:27:35Z","cross_cats_sorted":[],"title_canon_sha256":"860fa5944adaf638c6728643f16126cb55516e9a67eb779aa0004ac7483f9fa8","abstract_canon_sha256":"54155c0ce85ffc385f8347104437a2f9e2262a9dd07024790426d09bf77f1d6e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:48:33.733757Z","signature_b64":"XNquz1Eokr7KO/k6vS8Qni0QcGwsY6F71YqWaFp6Hw0fi9YqgQfcfqLFw8ChdS+ANPobbq2HHm2eJWhyC60iDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1f30145d419e29a57c4e9518ff1148fb0dc3ae6821ee1e1752b8bf5c5990ad7d","last_reissued_at":"2026-05-18T03:48:33.733364Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:48:33.733364Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the pointwise convergence of the sequence of partial Fourier Sums along lacunary subsequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Victor Lie","submitted_at":"2012-04-30T03:27:35Z","abstract_excerpt":"In his 2006 ICM invited address, Konyagin mentioned the following conjecture: if $S_n f$ stands for the $n$-th partial Fourier sum of $f$ and ${n_j}_j\\subset \\N$ is a lacunary sequence, then $S_{n_j} f$ is a.e. pointwise convergent for any $f\\in L\\log\\log L$. In this paper we will show that $| \\sup_{j} |S_{n_j}(f)| |_{1,\\infty}\\leq C |f|_{1} \\log\\log (10+\\frac{|f|_{\\infty}}{|f|_1})\\:.$ As a direct consequence we obtain that $S_{n_j}f \\rightarrow f $ a.e. for $f\\in L\\log\\log L\\log\\log\\log L$. 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