{"paper":{"title":"The cardinality of orthogonal exponentials of planar self-affine measures with three-element digit sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jing-Cheng Liu, Ming-Liang Chen","submitted_at":"2018-05-25T00:56:31Z","abstract_excerpt":"In this paper, we consider the planar self-affine measures $\\mu_{M,D}$ generated by an expanding matrix $M\\in M_2(\\mathbb{Z})$ and an integer digit set $\nD=\\left\\{ {\\left( {\\begin{array}{*{20}{c}} 0\\\\ 0\n\\end{array}} \\right),\\left( {\\begin{array}{*{20}{c}} \\alpha_1\\\\ \\alpha_2 \\end{array}} \\right),\\left( {\\begin{array}{*{20}{c}} \\beta_1\\\\ \\beta_2 \\end{array}} \\right)} \\right\\} $\nwith $\\alpha_1\\beta_2-\\alpha_2\\beta_1\\neq0$. We show that if $\\det(M)\\notin 3\\mathbb{Z}$, then the mutually orthogonal exponential functions in $L^2(\\mu_{M,D})$ is finite, and the exact maximal cardinality is given."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.09941","kind":"arxiv","version":1},"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"}