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

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.