{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T15:29:09Z","timestamp":1648654149396},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2013,4,5]],"date-time":"2013-04-05T00:00:00Z","timestamp":1365120000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2013,5]]},"abstract":"<jats:p>In a paper published in this journal, Alon, Kohayakawa, Mauduit, Moreira and R\u00f6dl proved that the minimal possible value of the normality measure of an <jats:italic>N<\/jats:italic>-element binary sequence satisfies\n<jats:disp-formula-group><jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0963548313000084_eqnU1\" \/><jats:tex-math>\n\\begin{equation*}\n\\biggl( \\frac{1}{2} + o(1) \\biggr) \\log_2 N \\leq \\min_{E_N \\in \\{0,1\\}^N} \\mathcal{N}(E_N) \\leq 3 N^{1\/3} (\\log N)^{2\/3}\n\\end{equation*}\n<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula><\/jats:disp-formula-group>\nfor sufficiently large <jats:italic>N<\/jats:italic>, and conjectured that the lower bound can be improved to some power of <jats:italic>N<\/jats:italic>. In this note it is observed that a construction of Levin of a normal number having small discrepancy gives a construction of a binary sequence <jats:italic>E<\/jats:italic><jats:sub><jats:italic>N<\/jats:italic><\/jats:sub> with <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548313000084_char1\" \/><\/jats:private-char>(<jats:italic>E<\/jats:italic><jats:sub><jats:italic>N<\/jats:italic><\/jats:sub>) = <jats:italic>O<\/jats:italic>((log <jats:italic>N<\/jats:italic>)<jats:sup>2<\/jats:sup>), thus disproving the conjecture above.<\/jats:p>","DOI":"10.1017\/s0963548313000084","type":"journal-article","created":{"date-parts":[[2013,4,5]],"date-time":"2013-04-05T06:50:44Z","timestamp":1365144644000},"page":"342-345","source":"Crossref","is-referenced-by-count":1,"title":["Normal Numbers and the Normality Measure"],"prefix":"10.1017","volume":"22","author":[{"given":"CHRISTOPH","family":"AISTLEITNER","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2013,4,5]]},"reference":[{"key":"S0963548313000084_ref11","unstructured":"Wall D. D. (1949) Normal numbers. PhD thesis, University of California, Berkeley."},{"key":"S0963548313000084_ref9","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970081"},{"key":"S0963548313000084_ref8","doi-asserted-by":"publisher","DOI":"10.4064\/aa-82-4-365-377"},{"key":"S0963548313000084_ref7","doi-asserted-by":"publisher","DOI":"10.4064\/aa-88-2-99-111"},{"key":"S0963548313000084_ref6","first-page":"361","article-title":"Numbers with bounded quotient and their applications to questions of Diophantine approximation","volume":"19","author":"Korobov","year":"1955","journal-title":"Izv. Akad. Nauk SSSR Ser. Mat."},{"key":"S0963548313000084_ref4","doi-asserted-by":"publisher","DOI":"10.1080\/10586458.2002.10504704"},{"key":"S0963548313000084_ref2","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548305007170"},{"key":"S0963548313000084_ref1","unstructured":"Aistleitner C. On the limit distribution of the normality measure of random binary sequences. Preprint. http:\/\/arxiv.org\/abs\/1301.6454."},{"key":"S0963548313000084_ref3","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/pdm027"},{"key":"S0963548313000084_ref5","volume-title":"The Art of Computer Programming","author":"Knuth","year":"1981"},{"key":"S0963548313000084_ref10","doi-asserted-by":"publisher","DOI":"10.4064\/aa-21-1-45-50"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548313000084","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,23]],"date-time":"2019-04-23T18:01:10Z","timestamp":1556042470000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548313000084\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,4,5]]},"references-count":11,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2013,5]]}},"alternative-id":["S0963548313000084"],"URL":"https:\/\/doi.org\/10.1017\/s0963548313000084","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,4,5]]}}}