{"paper":{"title":"Block-equivalent finite Gabor frames","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Finite Gabor systems are block-equivalent when either the modulation set or the translation set is a subgroup of Z_N.","cross_cats":[],"primary_cat":"math.FA","authors_text":"Laura De Carli, Luis Rodriguez, Oleg Asipchuk","submitted_at":"2026-05-15T16:19:19Z","abstract_excerpt":"We study finite systems of vectors whose frame operator matrices are unitarily equivalent, via explicit and computationally efficient unitary transformations, to block-diagonal matrices. We call such systems block-equivalent.\n  We show that a Gabor system $\\mathcal{G}=\\mathcal{G}(g,L\\times K)\\subset \\mathbb C^N$ is block-equivalent when either the modulation set $L$ or the translation set $K$ is a subgroup of $\\mathbb Z_N$. We also characterize situations in which the frame operator matrix becomes diagonal.\n  Finally, we show that geometric conditions on subsets of $\\mathbb Z_N$ force certain "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that a Gabor system G=G(g,L×K)⊂C^N is block-equivalent when either the modulation set L or the translation set K is a subgroup of Z_N. We also characterize situations in which the frame operator matrix becomes diagonal.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The unitary transformations achieving the block-diagonal form are both explicit and computationally efficient, as required by the definition of block-equivalence; this premise enters when the authors invoke the subgroup property to construct the equivalence.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Gabor systems in C^N with subgroup modulation or translation sets have frame operators that are unitarily equivalent to block-diagonal matrices, with further diagonal and sparsity results under geometric conditions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Finite Gabor systems are block-equivalent when either the modulation set or the translation set is a subgroup of Z_N.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0134caccb861ea9e6e743ed71b31db22f8ed5f54bccbbac9feb3c1ed5ad06561"},"source":{"id":"2605.16139","kind":"arxiv","version":1},"verdict":{"id":"fd878368-8af6-4baa-be44-05cd8df16359","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:28:20.643246Z","strongest_claim":"We show that a Gabor system G=G(g,L×K)⊂C^N is block-equivalent when either the modulation set L or the translation set K is a subgroup of Z_N. We also characterize situations in which the frame operator matrix becomes diagonal.","one_line_summary":"Gabor systems in C^N with subgroup modulation or translation sets have frame operators that are unitarily equivalent to block-diagonal matrices, with further diagonal and sparsity results under geometric conditions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The unitary transformations achieving the block-diagonal form are both explicit and computationally efficient, as required by the definition of block-equivalence; this premise enters when the authors invoke the subgroup property to construct the equivalence.","pith_extraction_headline":"Finite Gabor systems are block-equivalent when either the modulation set or the translation set is a subgroup of Z_N."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16139/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T19:01:18.949368Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T18:41:03.367890Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T18:22:03.939625Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:31.050224Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T16:41:55.454025Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"2b4f5cb74c881148f5fd61c09becd3711088ade89c52b38e9117984a5d163199"},"references":{"count":23,"sample":[{"doi":"","year":2024,"title":"L. 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Fickus,Finite normalized tight frames, Advances in Compu- tational Mathematics, vol. 18, no. 2–4, 2003, pp. 357–385","work_id":"ea5f7fcc-ec98-469c-a61f-f7c462097a29","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"Christensen,An Introduction to Frames and Riesz Bases, 2nd edition, Birkh¨ auser, Boston, 2016","work_id":"ed5d58ed-ac55-463a-a4e0-0af279ef21ad","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1999,"title":"E. M. Coven and A. Meyerowitz,Tiling the integers with translates of one finite set, Journal of Algebra, vol. 212, no. 1, 1999, pp. 161–174","work_id":"33bd8f83-830d-4699-bd6e-0cd0a1a439c3","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":23,"snapshot_sha256":"c75f6238312277eca277b93cd3f67eb0f379b457d384c873edda738ed2ffcac5","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"}