Measures associated with certain ellipsephic harmonic series and the Allouche-Hu-Morin limit theorem

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 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.

v2 January 1st, 2025. The ms has since been accepted for publication in Acta Mathematica Hungarica. Differs from v1 May 6, 2024 via a new title, some minor notational changes and a few bibliographical updates and other improvements

References in corpus (1)

• Summing the "exactly one 42" and similar subsums of the harmonic series