Fewest repetitions in infinite binary words

TL;DR

This paper constructs an infinite binary word with finitely many squares and no factors exceeding an exponent of 7/3, establishing the minimal number of squares needed and introducing the concept of finite-repetition threshold.

Contribution

It proves the existence of a binary infinite word with finitely many squares and bounded exponent, and identifies the minimal number of squares as 12, defining the finite-repetition threshold.

Findings

The infinite binary word contains exactly 12 squares.

It has exactly 2 factors with exponent 7/3.

These are the only factors with exponent larger than 2.

Abstract

A square is the concatenation of a nonempty word with itself. A word has period p if its letters at distance p match. The exponent of a nonempty word is the quotient of its length over its smallest period. In this article we give a proof of the fact that there exists an infinite binary word which contains finitely many squares and simultaneously avoids words of exponent larger than 7/3. Our infinite word contains 12 squares, which is the smallest possible number of squares to get the property, and 2 factors of exponent 7/3. These are the only factors of exponent larger than 2. The value 7/3 introduces what we call the finite-repetition threshold of the binary alphabet. We conjecture it is 7/4 for the ternary alphabet, like its repetitive threshold.