Approximations of additive squares in infinite words

TL;DR

This paper proves that infinite words over finite integer sets inevitably contain large segments resembling additive squares and segments with equal averages, revealing inherent additive structures.

Contribution

It introduces new results on the unavoidable presence of approximate additive squares and equal-average segments in infinite words over finite integer sets.

Findings

Infinite words contain arbitrarily large factors close to additive squares.

Existence of factors with segments sharing the same average for all k>1.

Demonstrates inherent additive patterns in infinite words.

Abstract

We show that every infinite word $\omega$ on a finite subset of $\mathbb{Z}$ must contain arbitrarily large factors $B_1B_2$ which are "close" to being \textit{additive squares}. We also show that for all $k>1, \ \omega$ must contain a factor $U_1U_2 ... U_k$ where $U_1,U_2,..., U_k$ all have the same \textit{average.}