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Kolountzakis","submitted_at":"2011-08-29T18:22:09Z","abstract_excerpt":"A bounded measurable set $\\Omega$, of Lebesgue measure 1, in the real line is called spectral if there is a set $\\Lambda$ of real numbers (\"frequencies\") such that the exponential functions $e_\\lambda(x) = \\exp(2\\pi i \\lambda x)$, $\\lambda\\in\\Lambda$, form a complete orthonormal system of $L^2(\\Omega)$. Such a set $\\Lambda$ is called a {\\em spectrum} of $\\Omega$. In this note we prove that any spectrum $\\Lambda$ of a bounded measurable set $\\Omega\\subseteq\\RR$ must be periodic."},"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":"1108.5689","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-08-29T18:22:09Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"ba3ac7d4b5665dbe162bea28c190c6858ac705fa4f9066303c967324c0102a38","abstract_canon_sha256":"ee70606c449b8e6f705a4b89522ca0444d72d9578f81ac8125c285915bd6cf0c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:01:50.938306Z","signature_b64":"gNDEqQisIUbPU2aBVwrTbgeZ3U556Ntds+kwCtIuyHHmRdLCUfDAtZxTwYerso7t24cJUPGczgKeAMTNEwT/CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a95e52b290083f5d9adcc361118bba0e50d500a80606e9e12519ab03625d8e9d","last_reissued_at":"2026-05-18T04:01:50.937756Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:01:50.937756Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Periodicity of the spectrum in dimension one","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Alex Iosevich, Mihail N. 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