On the Partial Sum of Subword-Counting Sequences

Pranjal Jain; Shuo Li

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arXiv · 2411.09819

On the Partial Sum of Subword-Counting Sequences

Pranjal Jain·Shuo Li

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Abstract

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 $\epsilon >0$.

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Pranjal Jain, Shuo Li. 2024-11-14. On the Partial Sum of Subword-Counting Sequences. https://arxiv.org/abs/2411.09819

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