{"paper":{"title":"On the topology of sums in powers of an algebraic number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Boris Solomyak, Nikita Sidorov","submitted_at":"2009-09-17T21:42:37Z","abstract_excerpt":"Let $1<q<2$ and \\[ \\Lambda(q)={\\sum_{k=0}^n a_kq^k\\mid a_k\\in\\{-1,0,1\\}, n\\ge1}. \\] It is well known that if $q$ is not a root of a polynomial with coefficients $0,\\pm1$, then $\\Lambda(q)$ is dense in $\\mathbb{R}$. We give several sufficient conditions for the denseness of $\\Lambda(q)$ when $q$ is a root of such a polynomial. In particular, we prove that if $q$ is not a Perron number or it has a conjugate $\\alpha$ such that $q|\\alpha|<1$, then $\\Lambda(q)$ is dense in $\\mathbb{R}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.3324","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"}