# On the longest common subsequence of Thue-Morse words

## TL;DR

This paper investigates the length of the longest common subsequence between Thue-Morse words and their complements, establishing that this length approaches the full sequence length as the sequence grows large.

## Contribution

The paper provides new lower bounds on the longest common subsequence length, demonstrating it approaches the sequence length asymptotically, and generalizes results to any prefix of the Thue-Morse sequence.

## Key findings

- Longest common subsequence length approaches the sequence length as n increases
- Constructed explicit common subsequences for lower bounds
- Generalized bounds to any prefix of Thue-Morse sequence

## Abstract

The length $a(n)$ of the longest common subsequence of the $n$'th Thue-Morse word and its bitwise complement is studied. An open problem suggested by Jean Berstel in 2006 is to find a formula for $a(n)$. In this paper we prove new lower bounds on $a(n)$ by explicitly constructing a common subsequence between the Thue-Morse words and their bitwise complement. We obtain the lower bound $a(n) = 2^{n}(1-o(1))$, saying that when $n$ grows large, the fraction of omitted symbols in the longest common subsequence of the $n$'th Thue-Morse word and its bitwise complement goes to $0$. We further generalize to any prefix of the Thue-Morse sequence, where we prove similar lower bounds.