{"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 $L^2(\\mu_{M,D})$. In particular, if $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.01250","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"}