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

TL;DR

This paper studies harmonic series over integers with specific digit patterns, relates them to measures on [0,1), and provides a new proof and error estimates for a related limit theorem involving digit blocks.

Contribution

It introduces a new measure-theoretic approach to analyze harmonic series with digit pattern constraints and offers a novel proof of the Allouche-Hu-Morin limit theorem.

Findings

Measures converge weakly to scaled Lebesgue measure

Provides a quantitative error estimate for the limit

Connects combinatorial generating functions to harmonic series analysis

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.