Let $w=x_0 x_1 x_2 \ldots$ be an infinite word, where each $x_i\in \{0,1\}$. For each positive integer $k$ (thought of as the jump size of an arithmetic progression) and each residue $0\leq a \leq k-1$ we can form the new "arithmetic progression" word $w_{a\ {\rm mod}\ k}=x_a x_{a+k} x_{a+2k}\ldots$.

**Question 1:** Does there exist a word $w$, as above, so that each of the derived words $w_{a\ {\rm mod}\ k}$ has no arbitrarily long constant subwords?

I tried finding the answer, but failed. I did find in the literature that for non-binary languages it is possible to do this (and more) without any repetitions at all!

I believe that the answer to Question 1 is probably well-known to experts, and it is positive. If so, I'm interested in the following extension.

**Question 2:** Can the upper bound on the length of the constant subwords of $w_{a\ {\rm mod}\ k}$ be made *independent* of $a$ and $k$? (If so, what is the smallest such bound?)