{"paper":{"title":"The Tu--Deng Conjecture holds almost surely","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CR","math.NT"],"primary_cat":"math.CO","authors_text":"Lukas Spiegelhofer, Michael Wallner","submitted_at":"2017-07-25T12:12:35Z","abstract_excerpt":"The Tu--Deng Conjecture is concerned with the sum of digits $w(n)$ of $n$ in base~$2$ (the Hamming weight of the binary expansion of $n$) and states the following: assume that $k$ is a positive integer and $1\\leq t<2^k-1$. Then \\[\\Bigl \\lvert\\Bigl\\{(a,b)\\in\\bigl\\{0,\\ldots,2^k-2\\bigr\\}^2:a+b\\equiv t\\bmod 2^k-1, w(a)+w(b)<k\\Bigr\\}\\Bigr \\rvert\\leq 2^{k-1}.\\]\n  We prove that the Tu--Deng Conjecture holds almost surely in the following sense: the proportion of $t\\in[1,2^k-2]$ such that the above inequality holds approaches $1$ as $k\\rightarrow\\infty$.\n  Moreover, we prove that the Tu--Deng Conjectu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.07945","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"}