Expansions of generalized Thue-Morse numbers

Abstract

We introduce generalized Thue-Morse numbers of the formπβ(θ):=∑n=1∞θnβn where β∈(1,m+1] with m∈N and θ=(θn)n≥1∈{0,1,⋯,m}N is a generalized Thue-Morse sequence previously studied by many authors in different terms. This is a natural generalization of the classical Thue-Morse number ∑n=1∞tn2n where (tn)n≥0 is the well-known Thue-Morse sequence 01101001⋯. We study when θ would be the unique, greedy, lazy, quasi-greedy and quasi-lazy β-expansions of πβ(θ), and generalize a result given by Kong and Li in 2015. In particular we deduce that the shifted Thue-Morse sequence (tn)n≥1 is the unique β-expansion of ∑n=1∞tnβn if and only if it is the greedy expansion, if and only if it is the lazy expansion, if and only if it is the quasi-greedy expansion, if and only if it is the quasi-lazy expansion, and if and only if β is no less than the Komornik-Loreti constant.