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 $\infty$.
Categories: math.NT