{"paper":{"title":"An equivalent of Kronecker's Theorem for powers of an Algebraic Number and Structure of Linear Recurrences of fixed length","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Maurizio Monge, Nevio Dubbini","submitted_at":"2009-10-27T17:21:47Z","abstract_excerpt":"After defining a notion of $\\epsilon$-density, we provide for any real algebraic number $\\alpha$ an estimate of the smallest $\\epsilon$ such that for each $m>1$ the set of vectors of the form $(t,t\\alpha,...,t\\alpha^{m-1})$ for $t\\in\\R$ is $\\epsilon$-dense modulo 1, in terms of the multiplicative Mahler measure $M(A(x))$ of the minimal integral polynomial $A(x)$ of $\\alpha$, and independently of $m$. In particular, we show that if $\\alpha$ has degree $d$ it is possible to take $\\epsilon = 2^{[d/2]}/M(A(x))$.\n  On the other hand using asymptotic estimates for Toeplitz determinants we show that "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.5182","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"}