Approximations of additive squares in infinite words

Abstract

We show that every infinite word $\ddot{I}‰$ on a finite subset of $\mathbb{Z}$ must contain arbitrarily large factors $B_1{B}_2$ which are "close" to being \textit{additive squares}. We also show that for all $k>1,\ddot{I}‰$ must contain a factor $U_1{U}_2...U_k$ where $U_1,U_2,...,U_k$ all have the same \textit{average.}