On the Partial Sum of Subword-Counting Sequences

Let $w$ be a finite word over the alphabet $\{0,1\}$. For any natural number $n$, let $s_w(n)$ denote the number of occurrence of $w$ in the binary expansion of $n$ as a scattered subsequence. We study the behavior of the partial sum $\sum_{n=0}^N(-1)^{s_w(n)}$ and characterize several classes of words $w$ satisfying $\sum_{n=0}^N(-1)^{s_w(n)}= O(N^{1-\epsilon})$ for some $\epsilon >0$.