  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 $\infty$.
