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Let $p\\geq2$ be a positive integer, $E_p^2:=\\frac{1}{p}\\{(i,j)^t:0\\leq i,j\\leq p-1\\}$ and $\\mathcal{Z}_{D}^2:=\\{x\\in[0, 1)^2:\\sum_{d\\in D}{e^{2\\pi i\\langle d,x\\rangle}}=0\\}$. We show that if $\\emptyset\\neq\\mathcal{Z}_{D}^2\\subset E_p^2\\setminus\\{0\\}$ and $\\gcd(\\det(M),p)=1$, then there exist at most $p^2$ mutually orthogonal exponential functions in $L^2(\\mu_{M,D})$. In particular, if $"},"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":"1611.01250","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-11-04T02:44:01Z","cross_cats_sorted":[],"title_canon_sha256":"fbe66eb90efcbc764255a8eaf5cca6bc229b62bb21a39af455c1f39f883c729a","abstract_canon_sha256":"9f12a9230e5a9a7e1469e317490000b28f73a9e76883603da33d7a0d6de379db"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:33.823715Z","signature_b64":"UqdZ8T7efUKN4Ryk8Qt7Jn0T1H1hjPUNcZrpmwMHXROWLdOCAPQ/mQli0ITQLtEIoxGqj1asX0ZzC56X0CSYBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a4096bd88e1f0090045480d8dd166d82235b28c97f23f8d21eddba566a71c25","last_reissued_at":"2026-05-18T00:51:33.823190Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:33.823190Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Non-spectral problem for the planar self-affine measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jian-lin Li, Jing-Cheng Liu, Xin-han Dong","submitted_at":"2016-11-04T02:44:01Z","abstract_excerpt":"In this paper, we consider the non-spectral problem for the planar self-affine measures $\\mu_{M,D}$ generated by an expanding integer matrix $M\\in M_2(\\mathbb{Z})$ and a finite digit set $D\\subset\\mathbb{Z}^2$. Let $p\\geq2$ be a positive integer, $E_p^2:=\\frac{1}{p}\\{(i,j)^t:0\\leq i,j\\leq p-1\\}$ and $\\mathcal{Z}_{D}^2:=\\{x\\in[0, 1)^2:\\sum_{d\\in D}{e^{2\\pi i\\langle d,x\\rangle}}=0\\}$. We show that if $\\emptyset\\neq\\mathcal{Z}_{D}^2\\subset E_p^2\\setminus\\{0\\}$ and $\\gcd(\\det(M),p)=1$, then there exist at most $p^2$ mutually orthogonal exponential functions in $L^2(\\mu_{M,D})$. 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