{"paper":{"title":"Approaching Cusick's conjecture on the sum-of-digits function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Lukas Spiegelhofer","submitted_at":"2019-04-18T09:14:17Z","abstract_excerpt":"Cusick's conjecture on the binary sum of digits $s(n)$ of a nonnegative integer $n$ states the following: for all nonnegative integers $t$ we have \\[ c_t=\\lim_{N\\rightarrow\\infty}\\frac 1N\\left\\lvert\\{n<N:s(n+t)\\geq s(n)\\}\\right\\rvert>1/2. \\] We prove that for given $\\varepsilon>0$ we have \\[ c_t+c_{t'}>1-\\varepsilon \\] if the binary expansion of $t$ contains enough blocks of consecutive $\\mathtt 1$s (depending on $\\varepsilon$), where $t'=3\\cdot 2^\\lambda-t$ and $\\lambda$ is chosen such that $2^\\lambda\\leq t<2^{\\lambda+1}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.08646","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"}