Avoiding Three Consecutive Blocks of the Same Size and Same Sum

TL;DR

This paper proves the existence of an infinite word over a four-letter alphabet that avoids three consecutive blocks of the same size and sum, solving a problem posed in 1994.

Contribution

It demonstrates the construction of such an infinite word, resolving an open problem in combinatorics on words.

Findings

Existence of an infinite word over {0,1,3,4} avoiding three consecutive identical blocks.

Provides a constructive proof addressing an open problem from 1994.

Advances understanding of pattern avoidance in infinite words.

Abstract

We show that there exists an infinite word over the alphabet {0, 1, 3, 4} containing no three consecutive blocks of the same size and the same sum. This answers an open problem of Pirillo and Varricchio from 1994.