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 $\epsilon >0$.