On the critical exponent of generalized Thue-Morse words

TL;DR

This paper investigates the critical exponent of generalized Thue-Morse words, calculating the supremum of rational exponents of powers within these words and identifying the specific occurrences that realize this exponent.

Contribution

It provides an exact computation of the critical exponent for generalized Thue-Morse words and characterizes the occurrences of powers that attain this exponent.

Findings

Determined the critical exponent for generalized Thue-Morse words.

Identified the specific positions where powers realizing the critical exponent occur.

Provided exact values for the supremum of rational exponents in these words.

Abstract

For certain generalized Thue-Morse words t, we compute the "critical exponent", i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.