The sum-of-digits function on arithmetic progressions

Let $s_2$ be the sum-of-digits function in base $2$, which returns the number of non-zero binary digits of a nonnegative integer $n$. We study $s_2$ alon g arithmetic subsequences and show that — up to a shift — the set of $m$-tuples of integers that appear as an arithmetic subsequence of $s_2$ has full complexity.