Approximations of additive squares in infinite words

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