From counting blocks to the Lebesgue measure, with an application to the Allouche-Hu-Morin limit theorem on block-constrained harmonic series

We consider the harmonic series S(k)=∑⁽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 bᵖ times the Lebesgue measure, a fact which allows a new proof of the theorem of Allouche, Hu, and Morin which says lim S(k)=bᵖlog(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.