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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"},"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":"1512.04234","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-12-14T09:45:10Z","cross_cats_sorted":["cs.FL"],"title_canon_sha256":"c1505c9553b09b8314b2cd23b28db5c224351049cff5ef846e08c2598d08239d","abstract_canon_sha256":"71831c5960ce41f9ac2fc482c6c12ec9ed4c3b7c9c364fd0de683a2fc528912c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:49.466866Z","signature_b64":"qbwy7K3eV4iYiUgxl0aY2Pou1m63RmLgMNlFJmy143BYlRhVRoN2XYRWANgcEmaSy9FebIE8nQRBrAQxw5EtDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0faa6dc92c4e48920124ab8b91dba2f5735a423f8a39b1533d5edadaad2cc6e1","last_reissued_at":"2026-05-18T00:20:49.466347Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:49.466347Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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