{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,29]],"date-time":"2025-11-29T16:12:24Z","timestamp":1764432744269,"version":"3.28.0"},"reference-count":8,"publisher":"Wiley","license":[{"start":{"date-parts":[[2010,5,1]],"date-time":"2010-05-01T00:00:00Z","timestamp":1272672000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>\u03b2<\/jats:italic>\u2208(1,2) be a Pisot number and let <jats:italic>H<\/jats:italic><jats:sub><jats:italic>\u03b2<\/jats:italic><\/jats:sub> denote Garsia\u2019s entropy for the Bernoulli convolution associated with <jats:italic>\u03b2<\/jats:italic>. Garsia, in 1963, showed that <jats:italic>H<\/jats:italic><jats:sub><jats:italic>\u03b2<\/jats:italic><\/jats:sub>&lt;1 for any Pisot <jats:italic>\u03b2<\/jats:italic>. For the Pisot numbers which satisfy <jats:italic>x<\/jats:italic><jats:sup><jats:italic>m<\/jats:italic><\/jats:sup>=<jats:italic>x<\/jats:italic><jats:sup><jats:italic>m<\/jats:italic>\u22121<\/jats:sup>+<jats:italic>x<\/jats:italic><jats:sup><jats:italic>m<\/jats:italic>\u22122<\/jats:sup>+\u22ef+<jats:italic>x<\/jats:italic>+1 (with <jats:italic>m<\/jats:italic>\u22652), Garsia\u2019s entropy has been evaluated with high precision by Alexander and Zagier for <jats:italic>m<\/jats:italic>=2 and later by Grabner, Kirschenhofer and Tichy for <jats:italic>m<\/jats:italic>\u22653, and it proves to be close to 1. No other numerical values for <jats:italic>H<\/jats:italic><jats:sub><jats:italic>\u03b2<\/jats:italic><\/jats:sub> are known. In the present paper we show that <jats:italic>H<\/jats:italic><jats:sub><jats:italic>\u03b2<\/jats:italic><\/jats:sub>&gt;0.81 for all Pisot <jats:italic>\u03b2<\/jats:italic>, and improve this lower bound for certain ranges of <jats:italic>\u03b2<\/jats:italic>. Our method is computational in nature.<\/jats:p>","DOI":"10.1112\/s1461157008000430","type":"journal-article","created":{"date-parts":[[2010,4,22]],"date-time":"2010-04-22T10:49:40Z","timestamp":1271933380000},"page":"130-143","source":"Crossref","is-referenced-by-count":5,"title":["A lower bound for Garsia\u2019s entropy for certain Bernoulli convolutions"],"prefix":"10.1112","volume":"13","author":[{"given":"Kevin G.","family":"Hare","sequence":"first","affiliation":[]},{"given":"Nikita","family":"Sidorov","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2010,5,1]]},"reference":[{"key":"S1461157008000430_ref22","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1090\/pspum\/072.1\/2112107","volume-title":"Fractal geometry and applications: a jubilee of Beno\u01d0t Mandelbrot. Part 1","author":"Solomyak","year":"2004"},{"key":"S1461157008000430_ref2","doi-asserted-by":"crossref","first-page":"215","DOI":"10.24033\/asens.1156","volume":"83","author":"Amara","year":"1966","journal-title":"Ann. Sci. \u00c9c. Norm. Sup\u00e9r. (3)"},{"key":"S1461157008000430_ref18","first-page":"280","volume":"223","author":"Pushkarev","year":"1995","journal-title":"Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)"},{"key":"S1461157008000430_ref23","first-page":"A739","volume":"287","author":"Talmoudi","year":"1978","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. A\u2013B"},{"key":"S1461157008000430_ref9","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.2140\/pjm.1963.13.1159","volume":"13","author":"Garsia","year":"1963","journal-title":"Pacific J. Math."},{"key":"S1461157008000430_ref15","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781107326019","volume-title":"Algebraic combinatorics on words","author":"Lothaire","year":"2002"},{"key":"S1461157008000430_ref13","doi-asserted-by":"crossref","first-page":"35","DOI":"10.24033\/bsmf.2411","volume":"130","author":"Komatsu","year":"2002","journal-title":"Bull. Soc. Math. France"},{"key":"S1461157008000430_ref17","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/978-3-0348-8380-1_2","volume-title":"Fractal geometry and stochastics, vol. II (Greifswald\/Koserow, 1998)","author":"Peres","year":"2000"}],"container-title":["LMS Journal of Computation and Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1461157008000430","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,20]],"date-time":"2024-11-20T13:15:15Z","timestamp":1732108515000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1461157008000430\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,5,1]]},"references-count":8,"alternative-id":["S1461157008000430"],"URL":"https:\/\/doi.org\/10.1112\/s1461157008000430","relation":{},"ISSN":["1461-1570"],"issn-type":[{"type":"electronic","value":"1461-1570"}],"subject":[],"published":{"date-parts":[[2010,5,1]]}}}