A regularity lemma and twins in words

TL;DR

This paper proves that any binary word of length n can be nearly evenly split into two disjoint identical subwords, using a regularity lemma for words, and extends the result to larger alphabets and multiple subwords.

Contribution

It introduces a regularity lemma for words and demonstrates that binary words can be almost perfectly partitioned into twins, advancing understanding of word structure and subword repetitions.

Findings

Binary words of length n contain nearly n/2 disjoint identical subwords.

The result extends to k identical subwords over alphabets with at most k letters.

The regularity lemma for words is a key tool in these proofs.

Abstract

For a word $S$, let $f(S)$ be the largest integer $m$ such that there are two disjoints identical (scattered) subwords of length $m$. Let $f(n, \Sigma) = \min \{f(S): S \text{is of length} n, \text{over alphabet} \Sigma \}$. Here, it is shown that \[2f(n, \{0,1\}) = n-o(n)\] using the regularity lemma for words. I.e., any binary word of length $n$ can be split into two identical subwords (referred to as twins) and, perhaps, a remaining subword of length $o(n)$. A similar result is proven for $k$ identical subwords of a word over an alphabet with at most $k$ letters.