Approximations of additive squares 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.}