{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T03:56:48Z","timestamp":1768708608299,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The\u00a0Tu\u2013Deng\u00a0Conjecture 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$1\\leqslant t&lt;2^k-1$. 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)&lt;k\\Bigr\\}\\Bigr \\rvert\\leqslant 2^{k-1}.\\]We prove that the\u00a0Tu\u2013Deng\u00a0Conjecture 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 $k\\rightarrow\\infty$.Moreover, we prove that the\u00a0Tu\u2013Deng\u00a0Conjecture implies a conjecture due to T. W. Cusick concerning the sum of digits of $n$ and $n+t$.<\/jats:p>","DOI":"10.37236\/7178","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T02:15:42Z","timestamp":1578622542000},"source":"Crossref","is-referenced-by-count":2,"title":["The Tu\u2013Deng Conjecture holds Almost Surely"],"prefix":"10.37236","volume":"26","author":[{"given":"Lukas","family":"Spiegelhofer","sequence":"first","affiliation":[]},{"given":"Michael","family":"Wallner","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2019,2,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i1p28\/7787","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i1p28\/7787","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:18:15Z","timestamp":1579216695000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i1p28"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,22]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2019,1,11]]}},"URL":"https:\/\/doi.org\/10.37236\/7178","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,2,22]]},"article-number":"P1.28"}}