{"title":"Measures associated with certain ellipsephic harmonic series and the Allouche–Hu–Morin limit theorem","authors":"J.-F. Burnol","doi":"10.1007/s10474-025-01525-3","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the harmonic series <span>\\(S(k)=\\sum^{(k)} m^{-1}\\)</span> over the integers having <span>\\(k\\)</span> occurrences of a given block of <span>\\(b\\)</span>-ary digits, of length <span>\\(p\\)</span>, and relate\nthem to certain measures on the interval [0, 1). We show that these measures converge weakly to <span>\\(b^p\\)</span> times the Lebesgue measure, a fact which allows a new proof\nof the theorem of Allouche, Hu, and Morin [4] which says <span>\\(\\lim S(k)=b^p\\log(b)\\)</span>.\nA quantitative error estimate will be given. Combinatorial aspects involve generating series which fall under the scope of the Goulden–Jackson cluster generating\nfunction formalism and the work of Guibas–Odlyzko on string overlaps.</p></div>","PeriodicalId":50894,"journal":{"name":"Acta Mathematica Hungarica","volume":"175 2","pages":"519 - 531"},"PeriodicalIF":0.6000,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Hungarica","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10474-025-01525-3","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the harmonic series \(S(k)=\sum^{(k)} m^{-1}\) over the integers having \(k\) occurrences of a given block of \(b\)-ary digits, of length \(p\), and relate
them to certain measures on the interval [0, 1). We show that these measures converge weakly to \(b^p\) times the Lebesgue measure, a fact which allows a new proof
of the theorem of Allouche, Hu, and Morin [4] which says \(\lim S(k)=b^p\log(b)\).
A quantitative error estimate will be given. Combinatorial aspects involve generating series which fall under the scope of the Goulden–Jackson cluster generating
function formalism and the work of Guibas–Odlyzko on string overlaps.
期刊介绍:
Acta Mathematica Hungarica is devoted to publishing research articles of top quality in all areas of pure and applied mathematics as well as in theoretical computer science. The journal is published yearly in three volumes (two issues per volume, in total 6 issues) in both print and electronic formats. Acta Mathematica Hungarica (formerly Acta Mathematica Academiae Scientiarum Hungaricae) was founded in 1950 by the Hungarian Academy of Sciences.