First, a bit of notation. If we have an arithmetic progression $a, a+k, a+2k, \ldots, a+(n-1)k$ we will call $k$ the distance, and $n$ the length.
While trying to find an example for a paper I'm writing in ring theory, I was led to ask the question: Is there a sequence of 0's and 1's for which if there is an arithmetic progression in the sequence which is constantly 0 (or 1), is there a bound for the length in terms of the distance?
I found that the answer is yes, and the Thue-Morse sequence works. Modifying the ideas of Corollary 2 in "Thue-Morse at multiples of an integer" (available here), we see that the length of a constant arithmetic progression on the Thue-Morse sequence of distance $k$ is bounded by $32k^3$.
So, here are my questions for you experts.
(1) Is there an easier sequence where one can prove this is true (possibly with citation in the literature)?
(2) If not, is there a straightforward citation for this fact for the Thue-Morse sequence? (The reference I gave above works for arithmetic progressions which start near the front of the sequence. But to get an arbitrary arithmetic progression, you need to increase the bound given in the paper a little, and also give a supplementary argument.)