{"paper":{"title":"Calculating the density of solutions of equations related to the P\\'olya-Ostrowski group through Markov chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.NT","authors_text":"Dario Spirito","submitted_at":"2018-03-12T14:28:24Z","abstract_excerpt":"Motivated by a problem in the theory of integer-valued polynomials, we investigate the natural density of the solutions of equations of the form $\\theta_uu_q(n)+\\theta_ww_q(n)+\\theta_2\\frac{n(n+1)}{2}+\\theta_1n+\\theta_0\\equiv 0\\bmod d$, where $d,q\\geq 2$ are fixed integers, $\\theta_u,\\theta_w,\\theta_2,\\theta_1,\\theta_0$ are parameters and $u_q$ and $w_q$ are functions related to the $q$-adic valuations of the numbers between 1 and $n$. We show that the number of solutions of this equation in $[0,N)$ satisfies a recurrence relation, with which we can associate to any pair $(d,q)$ a stochastic m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.04280","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"}