Spectrum of the refined Diophantine exponent

The refined Diophantine exponent, recently introduced by the author, is a quantity that measures the periodicity of an infinite word. In this article, we study this exponent from combinatorial and topological viewpoints. First, we show that, over a ternary alphabet, the spectrum of the refined Diophantine exponent is $[1,\mathrm{\infty }]$. Second, we show that this exponent has topological properties similar to those of the set of Liouville numbers. Finally, we provide concrete examples with the Champernowne, Rudin–Shapiro, and Thue–Morse words, words coming from coding a rotation by intervals, and bracket words.