The asymptotic repetition threshold of sequences rich in palindromes

TL;DR

This paper investigates the maximum repetition rate of factors in sequences rich in palindromes, establishing that for rich recurrent sequences over any alphabet size greater than one, the asymptotic repetition threshold is exactly two.

Contribution

It proves that the asymptotic repetition threshold for rich recurrent sequences over any alphabet size greater than one is exactly two, regardless of alphabet size.

Findings

Asymptotic repetition threshold for rich recurrent sequences is 2.

Threshold is independent of alphabet size for rich sequences.

Contrast with all d-ary sequences where the threshold is 1.

Abstract

The asymptotic critical exponent measures for a sequence the maximum repetition rate of factors of growing length. The infimum of asymptotic critical exponents of sequences of a certain class is called the asymptotic repetition threshold of that class. On the one hand, if we consider the class of all d-ary sequences with d greater than one, then the asymptotic repetition threshold is equal to one, independently of the alphabet size. On the other hand, for the class of episturmian sequences, the repetition threshold depends on the alphabet size. We focus on rich sequences, i.e., sequences whose factors contain the maximum possible number of distinct palindromes. The class of episturmian sequences forms a subclass of rich sequences. We prove that the asymptotic repetition threshold for the class of rich recurrent d-ary sequences, with d greater than one, is equal to two, independently of the alphabet size.