{"paper":{"title":"Two applications of the spectrum of numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.FL"],"primary_cat":"math.NT","authors_text":"Christiane Frougny, Edita Pelantov\\'a","submitted_at":"2015-12-14T09:45:10Z","abstract_excerpt":"Let the base $\\beta$ be a complex number, $|\\beta|>1$, and let $A \\subset \\C$ be a finite alphabet of digits. The \\emph{$A$-spectrum} of $\\beta$ is the set $S_{A}(\\beta) = \\{\\sum_{k=0}^n a_k\\beta^k \\mid n \\in \\mathbb{N}, \\ a_k \\in {A}\\}$. We show that the spectrum $S_{{A}}(\\beta)$ has an accumulation point if and only if $0$ has a particular $(\\beta, A)$-representation, said to be \\emph{rigid}.\n  The first application is restricted to the case that $\\beta >1 $ and the alphabet is $A=\\{-M, \\ldots, M\\}$, $M \\ge 1$ integer. We show that the set $Z_{\\beta,M}$ of infinite $(\\beta, A)$-representatio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04234","kind":"arxiv","version":3},"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"}