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We give sufficient conditions on $\\Lambda$, met by certain kinds of Cantor sets, for this to be a bounded sublinear operator. This result is a discrete analogue of E. M. Stein's integral result, that the maximal operator below is bounded on $L^2(\\mathbb R)$. \\[ \\mathcal{C}_2 f(x):= \\sup_{\\lambda \\in \\mathbb R} \\left| \\int f(x-y) \\frac{e^{2\\pi i \\lamb"},"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":"1512.06918","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-12-22T00:29:55Z","cross_cats_sorted":[],"title_canon_sha256":"7b25849a0f89dd22474029e743300915d59c88d844c68fd049bcbe683b801abc","abstract_canon_sha256":"59f7edf0ea2edeb6ca1bc13ad803e911916ec337110939f6f35cd00278d93f84"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:57.784103Z","signature_b64":"PDJ1sgZXN+hxJjY8TL6NytxNM/tUsGQTJv9Akhvg4D4Omr3BDwPzEXJyZDg1MEE6OUwureHQIS8wNvTif6kqDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0f1812e2705b14ffcad56a853ea838050e5450392ed2b2b47d28915b1f1e0701","last_reissued_at":"2026-05-18T01:15:57.783528Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:57.783528Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Discrete Quadratic Carleson Theorem on $ \\ell ^2 $ with a Restricted Supremum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ben Krause, Michael Lacey","submitted_at":"2015-12-22T00:29:55Z","abstract_excerpt":"Consider the discrete maximal function acting on $\\ell^2(\\mathbb Z)$ functions \\[ \\mathcal{C}_{\\Lambda} f( n ) := \\sup_{ \\lambda \\in \\Lambda} \\left| \\sum_{m \\neq 0} f(n-m) \\frac{e^{2 \\pi i\\lambda m^2}} {m} \\right| \\] where $\\Lambda \\subset [0,1]$. We give sufficient conditions on $\\Lambda$, met by certain kinds of Cantor sets, for this to be a bounded sublinear operator. This result is a discrete analogue of E. M. Stein's integral result, that the maximal operator below is bounded on $L^2(\\mathbb R)$. \\[ \\mathcal{C}_2 f(x):= \\sup_{\\lambda \\in \\mathbb R} \\left| \\int f(x-y) \\frac{e^{2\\pi i \\lamb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.06918","kind":"arxiv","version":3},"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":"1512.06918","created_at":"2026-05-18T01:15:57.783627+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.06918v3","created_at":"2026-05-18T01:15:57.783627+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.06918","created_at":"2026-05-18T01:15:57.783627+00:00"},{"alias_kind":"pith_short_12","alias_value":"B4MBFYTQLMKP","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_16","alias_value":"B4MBFYTQLMKP7SWV","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_8","alias_value":"B4MBFYTQ","created_at":"2026-05-18T12:29:14.074870+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/B4MBFYTQLMKP7SWVNKCT5KBYAU","json":"https://pith.science/pith/B4MBFYTQLMKP7SWVNKCT5KBYAU.json","graph_json":"https://pith.science/api/pith-number/B4MBFYTQLMKP7SWVNKCT5KBYAU/graph.json","events_json":"https://pith.science/api/pith-number/B4MBFYTQLMKP7SWVNKCT5KBYAU/events.json","paper":"https://pith.science/paper/B4MBFYTQ"},"agent_actions":{"view_html":"https://pith.science/pith/B4MBFYTQLMKP7SWVNKCT5KBYAU","download_json":"https://pith.science/pith/B4MBFYTQLMKP7SWVNKCT5KBYAU.json","view_paper":"https://pith.science/paper/B4MBFYTQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.06918&json=true","fetch_graph":"https://pith.science/api/pith-number/B4MBFYTQLMKP7SWVNKCT5KBYAU/graph.json","fetch_events":"https://pith.science/api/pith-number/B4MBFYTQLMKP7SWVNKCT5KBYAU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/B4MBFYTQLMKP7SWVNKCT5KBYAU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/B4MBFYTQLMKP7SWVNKCT5KBYAU/action/storage_attestation","attest_author":"https://pith.science/pith/B4MBFYTQLMKP7SWVNKCT5KBYAU/action/author_attestation","sign_citation":"https://pith.science/pith/B4MBFYTQLMKP7SWVNKCT5KBYAU/action/citation_signature","submit_replication":"https://pith.science/pith/B4MBFYTQLMKP7SWVNKCT5KBYAU/action/replication_record"}},"created_at":"2026-05-18T01:15:57.783627+00:00","updated_at":"2026-05-18T01:15:57.783627+00:00"}