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Clearly, the basis $\\{\\chi_{\\alpha,n}(x):=\\exp(2\\pi i \\alpha x/n)\\}$ depends on the value $n$.\n  We show that if $f$ has \"large\" Fourier coefficients, then the function $\\widetilde{f}:\\mathbb{Z}_m \\to \\mathbb{C}$, given by \\[\n  \\widetilde{f}(x)\n  = \\begin{cases} f(x) & \\text{when } 0\\leq x < \\min(n, m),\n  0 & \\text{otherwise},\n  \\end{cases} \\] also has \"large\" coefficient"},"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":"1610.04330","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-10-14T04:44:49Z","cross_cats_sorted":[],"title_canon_sha256":"3248bdb94cb0cb7e6fc9f34af2492b0a0d310d44fe984bb15c3222ddf907a618","abstract_canon_sha256":"412ef90934f9757255020bd0c18a24727473153f1d28947e9dfe9e408631c4dc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:15.305925Z","signature_b64":"76S1VQuxn9jIDc+woahAHYTwtJpBQPloNm1tll33iZ2GnAHn0fEi5Zpmz+EoPzZ6UobCSsT3y4zVBwU0BHfyDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ff9e84561c43ec982c7bcc3069f4d26d3a881d00c4a9b1e7c2f9beb438ee6735","last_reissued_at":"2026-05-18T01:01:15.305240Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:15.305240Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Sets of Large Fourier Transform Under Changes in Domain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Barak Shani, Joel Laity","submitted_at":"2016-10-14T04:44:49Z","abstract_excerpt":"A function $f:\\mathbb{Z}_n \\to \\mathbb{C}$ can be represented as a linear combination $f(x)=\\sum_{\\alpha \\in \\mathbb{Z}_n}\\widehat{f}(\\alpha) \\chi_{\\alpha,n}(x)$ where $\\widehat{f}$ is the (discrete) Fourier transform of $f$. Clearly, the basis $\\{\\chi_{\\alpha,n}(x):=\\exp(2\\pi i \\alpha x/n)\\}$ depends on the value $n$.\n  We show that if $f$ has \"large\" Fourier coefficients, then the function $\\widetilde{f}:\\mathbb{Z}_m \\to \\mathbb{C}$, given by \\[\n  \\widetilde{f}(x)\n  = \\begin{cases} f(x) & \\text{when } 0\\leq x < \\min(n, m),\n  0 & \\text{otherwise},\n  \\end{cases} \\] also has \"large\" coefficient"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.04330","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1610.04330","created_at":"2026-05-18T01:01:15.305366+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.04330v2","created_at":"2026-05-18T01:01:15.305366+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.04330","created_at":"2026-05-18T01:01:15.305366+00:00"},{"alias_kind":"pith_short_12","alias_value":"76PIIVQ4IPWJ","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_16","alias_value":"76PIIVQ4IPWJQLD3","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_8","alias_value":"76PIIVQ4","created_at":"2026-05-18T12:30:04.600751+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/76PIIVQ4IPWJQLD3ZQYGT5GSNU","json":"https://pith.science/pith/76PIIVQ4IPWJQLD3ZQYGT5GSNU.json","graph_json":"https://pith.science/api/pith-number/76PIIVQ4IPWJQLD3ZQYGT5GSNU/graph.json","events_json":"https://pith.science/api/pith-number/76PIIVQ4IPWJQLD3ZQYGT5GSNU/events.json","paper":"https://pith.science/paper/76PIIVQ4"},"agent_actions":{"view_html":"https://pith.science/pith/76PIIVQ4IPWJQLD3ZQYGT5GSNU","download_json":"https://pith.science/pith/76PIIVQ4IPWJQLD3ZQYGT5GSNU.json","view_paper":"https://pith.science/paper/76PIIVQ4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.04330&json=true","fetch_graph":"https://pith.science/api/pith-number/76PIIVQ4IPWJQLD3ZQYGT5GSNU/graph.json","fetch_events":"https://pith.science/api/pith-number/76PIIVQ4IPWJQLD3ZQYGT5GSNU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/76PIIVQ4IPWJQLD3ZQYGT5GSNU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/76PIIVQ4IPWJQLD3ZQYGT5GSNU/action/storage_attestation","attest_author":"https://pith.science/pith/76PIIVQ4IPWJQLD3ZQYGT5GSNU/action/author_attestation","sign_citation":"https://pith.science/pith/76PIIVQ4IPWJQLD3ZQYGT5GSNU/action/citation_signature","submit_replication":"https://pith.science/pith/76PIIVQ4IPWJQLD3ZQYGT5GSNU/action/replication_record"}},"created_at":"2026-05-18T01:01:15.305366+00:00","updated_at":"2026-05-18T01:01:15.305366+00:00"}