# Repetitions in infinite palindrome-rich words

## TL;DR

This paper investigates repetitions in infinite rich words, establishing lower bounds on repetition thresholds and constructing a specific example over a binary alphabet with a notably small critical exponent, advancing open problems in the field.

## Contribution

It provides new lower bounds on repetition thresholds and constructs an infinite rich word with a minimal critical exponent over a binary alphabet, addressing a 2017 open problem.

## Key findings

- Established lower bounds on repetition thresholds for rich words over 2 and 3-letter alphabets.
- Constructed an infinite rich word over binary alphabet with critical exponent $2+rac{\sqrt{2}}{2}$.
- First progress on Vesti's open problem from 2017.

## Abstract

Rich words are characterized by containing the maximum possible number of distinct palindromes. Several characteristic properties of rich words have been studied; yet the analysis of repetitions in rich words still involves some interesting open problems. We address lower bounds on the repetition threshold of infinite rich words over 2 and 3-letter alphabets, and construct a candidate infinite rich word over the alphabet $\Sigma_2=\{0,1\}$ with a small critical exponent of $2+\sqrt{2}/2$. This represents the first progress on an open problem of Vesti from 2017.