How many double squares can a string contain?

TL;DR

This paper investigates the maximum number of distinct squares in a string, improving the upper bound from previous results by analyzing the structure of double squares.

Contribution

It introduces a new upper bound of 5n/3 for the number of distinct squares and provides a structural analysis of double squares, offering a novel proof of earlier bounds.

Findings

A string of length n contains at most 5n/3 distinct squares.

A string of length n contains at most 2n/3 double squares.

The structural properties of double squares lead to a new proof of existing bounds.

Abstract

Counting the types of squares rather than their occurrences, we consider the problem of bounding the number of distinct squares in a string. Fraenkel and Simpson showed in 1998 that a string of length n contains at most 2n distinct squares. Ilie presented in 2007 an asymptotic upper bound of 2n - Theta(log n). We show that a string of length n contains at most 5n/3 distinct squares. This new upper bound is obtained by investigating the combinatorial structure of double squares and showing that a string of length n contains at most 2n/3 double squares. In addition, the established structural properties provide a novel proof of Fraenkel and Simpson's result.