On the behavior of binary block-counting functions under addition

TL;DR

This paper studies how the counts of specific binary digit blocks change under addition, showing that for certain numbers, these differences follow a Gaussian distribution, extending previous results on binary digit sum functions.

Contribution

The paper generalizes earlier work on binary digit sums to block-counting functions, demonstrating Gaussian behavior for differences when adding numbers with many binary ones.

Findings

Differences in block counts are approximately normally distributed.

Results extend previous findings from digit sums to block-counting functions.

Distribution closeness to Gaussian improves with more binary ones in the addend.

Abstract

Let $\mathsf{s}(n)$ denote the sum of binary digits of an integer $n \geq 0$. In the recent years there has been interest in the behavior of the differences $\mathsf{s}(n+t)-\mathsf{s}(n)$, where $t \geq 0$ is an integer. In particular, Spiegelhofer and Wallner showed that for $t$ whose binary expansion contains sufficiently many blocks of $\mathtt{1}$s the inequality $\mathsf{s}(n+t) -\mathsf{s}(n) \geq 0$ holds for $n$ belonging to a set of asymptotic density $>1/2$, partially answering a question by Cusick. Furthermore, for such $t$ the values $\mathsf{s}(n+t) - \mathsf{s}(n)$ are approximately normally distributed. In this paper we consider a natural generalization to the family of block-counting functions $N^w$, giving the number of occurrences of a block of binary digits $w$ in the binary expansion. Our main result show that for any $w$ of length at least $2$ the distribution of the differences $N^w(n+t) - N^w(n)$ is close to a Gaussian when $t$ contains many blocks of $\mathtt{1}$s in its binary expansion. This extends an earlier result by the author and Spiegelhofer for $w=\mathtt{11}$.