{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T18:13:02Z","timestamp":1771611182289,"version":"3.50.1"},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T00:00:00Z","timestamp":1699833600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>It is proven that a conjecture of Tao (2010) holds true for log-concave random variables on the integers: For every <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000408_inline1.png\"\/><jats:tex-math>\n$n \\geq 1$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000408_inline2.png\"\/><jats:tex-math>\n$X_1,\\ldots,X_n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> are i.i.d. integer-valued, log-concave random variables, then<jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0963548323000408_eqnU1.png\"\/><jats:tex-math>\n\\begin{equation*} H(X_1+\\cdots +X_{n+1}) \\geq H(X_1+\\cdots +X_{n}) + \\frac {1}{2}\\log {\\Bigl (\\frac {n+1}{n}\\Bigr )} - o(1) \\end{equation*}\n<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula>as <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000408_inline3.png\"\/><jats:tex-math>\n$H(X_1) \\to \\infty$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000408_inline4.png\"\/><jats:tex-math>\n$H(X_1)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> denotes the (discrete) Shannon entropy. The problem is reduced to the continuous setting by showing that if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000408_inline5.png\"\/><jats:tex-math>\n$U_1,\\ldots,U_n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> are independent continuous uniforms on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000408_inline6.png\"\/><jats:tex-math>\n$(0,1)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, then<jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0963548323000408_eqnU2.png\"\/><jats:tex-math>\n\\begin{equation*} h(X_1+\\cdots +X_n + U_1+\\cdots +U_n) = H(X_1+\\cdots +X_n) + o(1), \\end{equation*}\n<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula>as <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000408_inline7.png\"\/><jats:tex-math>\n$H(X_1) \\to \\infty$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000408_inline8.png\"\/><jats:tex-math>\n$h$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> stands for the differential entropy. Explicit bounds for the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548323000408_inline9.png\"\/><jats:tex-math>\n$o(1)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-terms are provided.<\/jats:p>","DOI":"10.1017\/s0963548323000408","type":"journal-article","created":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T09:17:23Z","timestamp":1699867043000},"page":"196-209","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":2,"title":["Approximate discrete entropy monotonicity for log-concave sums"],"prefix":"10.1017","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3011-9435","authenticated-orcid":false,"given":"Lampros","family":"Gavalakis","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,11,13]]},"reference":[{"key":"S0963548323000408_ref9","first-page":"93","article-title":"An entropy power inequality for the binomial family","volume":"4","author":"Harremo\u00e9s","year":"2003","journal-title":"JIPAM. J. Inequal. Pure Appl. Math."},{"key":"S0963548323000408_ref17","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548309990642"},{"key":"S0963548323000408_ref3","doi-asserted-by":"publisher","DOI":"10.1017\/S096354832100016X"},{"key":"S0963548323000408_ref12","doi-asserted-by":"publisher","DOI":"10.1137\/1124073"},{"key":"S0963548323000408_ref15","doi-asserted-by":"publisher","DOI":"10.1002\/j.1538-7305.1948.tb01338.x"},{"key":"S0963548323000408_ref11","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2007.899484"},{"key":"S0963548323000408_ref2","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176992632"},{"key":"S0963548323000408_ref18","unstructured":"[18] Tao, T. and Vu, V. H. (2005) Entropy methods [Online]. Available: http:\/\/www.math.ucla.edu\/~tao\/preprints\/Expository\/."},{"key":"S0963548323000408_ref16","doi-asserted-by":"publisher","DOI":"10.1016\/S0019-9958(59)90348-1"},{"key":"S0963548323000408_ref7","doi-asserted-by":"crossref","unstructured":"[7] Haghighatshoar, S. , Abbe, E. and Telatar, E. (2012) Adaptive sensing using deterministic partial Hadamard matrices. IEEE, pp. 1842\u20131846, In 2012 IEEE International Symposium on Information Theory Proceedings.","DOI":"10.1109\/ISIT.2012.6283598"},{"key":"S0963548323000408_ref20","doi-asserted-by":"publisher","DOI":"10.1109\/ISIT.2015.7282731"},{"key":"S0963548323000408_ref8","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2014.2317181"},{"key":"S0963548323000408_ref4","volume-title":"Elements of Information Theory","author":"Cover","year":"2006"},{"key":"S0963548323000408_ref19","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511755149"},{"key":"S0963548323000408_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/BF02218051"},{"key":"S0963548323000408_ref6","unstructured":"[6] Gavalakis, L. and Kontoyiannis, I. (2021) Entropy and the discrete central limit theorem, arXiv preprint arXiv: 2106.00514."},{"key":"S0963548323000408_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/BF00535889"},{"key":"S0963548323000408_ref1","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-04-00459-X"},{"key":"S0963548323000408_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(74)90071-9"},{"key":"S0963548323000408_ref14","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20248"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548323000408","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,5]],"date-time":"2024-02-05T11:22:55Z","timestamp":1707132175000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548323000408\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,13]]},"references-count":20,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,3]]}},"alternative-id":["S0963548323000408"],"URL":"https:\/\/doi.org\/10.1017\/s0963548323000408","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,13]]},"assertion":[{"value":"\u00a9 The Author(s), 2023. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}