{"paper":{"title":"Multiple lattice tiles and Riesz bases of exponentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Mihail N. Kolountzakis","submitted_at":"2013-05-12T21:01:10Z","abstract_excerpt":"Suppose $\\Omega\\subseteq\\RR^d$ is a bounded and measurable set and $\\Lambda \\subseteq \\RR^d$ is a lattice. Suppose also that $\\Omega$ tiles multiply, at level $k$, when translated at the locations $\\Lambda$. This means that the $\\Lambda$-translates of $\\Omega$ cover almost every point of $\\RR^d$ exactly $k$ times. We show here that there is a set of exponentials $\\exp(2\\pi i t\\cdot x)$, $t\\in T$, where $T$ is some countable subset of $\\RR^d$, which forms a Riesz basis of $L^2(\\Omega)$. This result was recently proved by Grepstad and Lev under the extra assumption that $\\Omega$ has boundary of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.2632","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"}