{"paper":{"title":"On the asymptotic behaviour of the correlation measure of sum-of-digits function in base 2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT","math.PR"],"primary_cat":"math.CO","authors_text":"(2) MIPT), Alexander Prikhod'Ko (2) ((1) AMU, I2M, Jordan Emme (1)","submitted_at":"2015-04-07T18:30:34Z","abstract_excerpt":"Let $s\\_2(x)$ denote the number of digits \"$1$\" in a binary expansion of any $x \\in \\mathbb{N}$. We study the mean distribution $\\mu\\_a$ of the quantity $s\\_2(x+a)-s\\_2(x)$ for a fixed positive integer $a$.It is shown that solutions of the equation$$ s\\_2(x+a)-s\\_2(x)= d $$are uniquely identified by a finite set of prefixes in $\\{0,1\\}^*$, and that the probability distribution of differences $d$ is given by an infinite product of matrices whose coefficients are operators of $l^1(\\mathbb{Z})$.Then, denoting by $l(a)$ the number of patterns \"$01$\" in the binary expansion of $a$, we give the asym"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.01701","kind":"arxiv","version":2},"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"}