Infinite words containing squares at every position

TL;DR

This paper determines the minimal exponent /3 for which an infinite binary word can avoid /-powers while containing arbitrarily large squares starting at every position.

Contribution

It solves a specific open problem by proving that /3 is the infimum for the existence of such infinite words over a binary alphabet.

Findings

The infimum /3 is the threshold for the described property.

No infinite binary word can avoid /3-powers and have large squares at every position below this exponent.

The result fully characterizes the boundary for this combinatorial property.

Abstract

Richomme asked the following question: what is the infimum of the real numbers $\alpha$ > 2 such that there exists an infinite word that avoids $\alpha$-powers but contains arbitrarily large squares beginning at every position? We resolve this question in the case of a binary alphabet by showing that the answer is $\alpha$ = 7/3.