{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:PR2WVD2U3NTYXYCWTWDIZ5BFTM","short_pith_number":"pith:PR2WVD2U","schema_version":"1.0","canonical_sha256":"7c756a8f54db678be0569d868cf4259b09a9ac23bcf7ec858e567ee22f6a0e94","source":{"kind":"arxiv","id":"1609.03435","version":2},"attestation_state":"computed","paper":{"title":"On the Erd\\\"{o}s flat polynomials problem, Chowla conjecture and Riemann Hypothesis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.NT"],"primary_cat":"math.CO","authors_text":"el Houcein el Abdalaoui","submitted_at":"2016-09-12T15:08:11Z","abstract_excerpt":"There are no square $L^2$-flat sequences of polynomials of the type\n  $$\\frac{1}{\\sqrt q}( \\epsilon_0 + \\epsilon_1z + \\epsilon_2z^2 + \\cdots + \\epsilon_{q-2}z^{q-2} +\\epsilon_q z^{q-1}),$$ where for each $j,~~ 0 \\leq j\\leq q-1,~\\epsilon_j = \\pm 1$. It follows that Erd\\\"{o}s's conjectures on Littlewood polynomials hold. Consequently, Turyn-Golay's conjecture is true, that is, there are only finitely many Barker sequences. We further get that the spectrum of dynamical systems arising from continuous Morse sequences is singular. This settles an old question due to M. Keane. Applying our reasoning"},"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":"1609.03435","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-12T15:08:11Z","cross_cats_sorted":["math.CV","math.NT"],"title_canon_sha256":"d5a5798b17eb9b23e28e94014c6421c2adb1f88665c3a87c98b85e7a5e0b4641","abstract_canon_sha256":"6d047d7fb3289d3c5dff541374b2fa4b1025adf821136ea2585a2f0fd535be73"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:02.300170Z","signature_b64":"obSiDZ/UT21OBiElDyl/qqkuPyGahcAXfDtmTjBcWXCj5kXuz5L79RVVWoc5fLinlLzSY7ibhVTtzpZTVrbLDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c756a8f54db678be0569d868cf4259b09a9ac23bcf7ec858e567ee22f6a0e94","last_reissued_at":"2026-05-18T00:53:02.299556Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:02.299556Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Erd\\\"{o}s flat polynomials problem, Chowla conjecture and Riemann Hypothesis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.NT"],"primary_cat":"math.CO","authors_text":"el Houcein el Abdalaoui","submitted_at":"2016-09-12T15:08:11Z","abstract_excerpt":"There are no square $L^2$-flat sequences of polynomials of the type\n  $$\\frac{1}{\\sqrt q}( \\epsilon_0 + \\epsilon_1z + \\epsilon_2z^2 + \\cdots + \\epsilon_{q-2}z^{q-2} +\\epsilon_q z^{q-1}),$$ where for each $j,~~ 0 \\leq j\\leq q-1,~\\epsilon_j = \\pm 1$. It follows that Erd\\\"{o}s's conjectures on Littlewood polynomials hold. Consequently, Turyn-Golay's conjecture is true, that is, there are only finitely many Barker sequences. We further get that the spectrum of dynamical systems arising from continuous Morse sequences is singular. This settles an old question due to M. Keane. 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