{"paper":{"title":"A hidden signal in Hofstadter's $H$ sequence","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.DS"],"primary_cat":"math.NT","authors_text":"Rodrigo Angelo","submitted_at":"2022-06-01T20:21:49Z","abstract_excerpt":"The Hofstadter $H$ sequence is defined by $H(1) = 1$ and $H(n) = n-H(H(H(n-1)))$ for $n > 1$. If $\\alpha$ is the real root of $x^3+x=1$ we show that the numbers $\\alpha H(n) \\mod 1$ are not uniformly distributed on $[0,1]$, but converge to a distribution we believe is continuous but not differentiable. This is motivated by a discovery of Steinerberger, who found a real number with similar behavior for the Ulam sequence. Our result is related with the fact that a certain sequence defined from the linear recurrence $h_n=h_{n-1}+h_{n-3}$ has the property $\\|x h_n\\| \\rightarrow 0$ precisely for $x"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.00750","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2206.00750/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}