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

2024/05/06 by Burnol, Jean-François
#05A15 (Primary) 11A63 #11Y60 #28A25 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2405.03625

Abstract

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 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^plog(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.

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