Block occurrences in the binary expansion

Abstract

The binary sum-of-digits function $\mathsf{s}$ returns the number of ones in the binary expansion of a nonnegative integer. Cusick's Hamming weight conjecture states that, for all integers $t\geq 0$, the set of nonnegative integers $n$ such that $\mathsf{s}(n+t)\geq \mathsf{s}(n)$ has asymptotic density strictly larger than $1/2$. We are concerned with the block-additive function $\mathsf{r}$ returning the number of (overlapping) occurrences of the block $\mathtt{11}$ in the binary expansion of $n$. The main result of this paper is a central limit-type theorem for the difference $\mathsf{r}(n+t)-\mathsf{r}(n)$: the corresponding probability function is uniformly close to a Gaussian, where the uniform error tends to $0$ as the number of blocks of ones in the binary expansion of $t$ tends to $\infty$.