Van der Waerden's Theorem and Avoidability in Words

TL;DR

This paper explores the existence of infinite words avoiding certain repeated sum patterns, connecting the problem to van der Waerden's theorem on arithmetic progressions, and investigates variations of this problem.

Contribution

It introduces new variations of the avoidability problem in words, linking it to van der Waerden's theorem, and advances understanding of pattern avoidance in infinite words.

Findings

Identifies conditions for avoiding repeated sum blocks in infinite words

Establishes connections between pattern avoidance and arithmetic progressions

Provides new insights into the structure of words avoiding certain patterns

Abstract

Pirillo and Varricchio, and independently, Halbeisen and Hungerbuhler considered the following problem, open since 1994: Does there exist an infinite word w over a finite subset of Z such that w contains no two consecutive blocks of the same length and sum? We consider some variations on this problem in the light of van der Waerden's theorem on arithmetic progressions.