The critical exponent of the Arshon words

TL;DR

This paper determines the critical exponent of Arshon words for all orders n ≥ 2, showing it equals (3n-2)/(2n-2) and is achieved at the first position, generalizing previous specific cases.

Contribution

It generalizes the calculation of the critical exponent for Arshon words to all n ≥ 2, extending earlier results for n=2 and n=3.

Findings

Critical exponent of Arshon words is (3n-2)/(2n-2) for all n ≥ 2.

The critical exponent is attained at position 1.

The result generalizes previous specific cases.

Abstract

Generalizing the results of Thue (for n = 2) and of Klepinin and Sukhanov (for n = 3), we prove that for all n greater than or equal to 2, the critical exponent of the Arshon word of order $n$ is given by (3n-2)/(2n-2), and this exponent is attained at position 1.