Block occurrences in the binary expansion

The binary sum-of-digits function 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≥0 the set of integers n≥0 such that s(n+t)≥s(n) has asymptotic density strictly larger than 1/2. So far, the strongest results are due to the second author and M. Wallner [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 24, No. 1, 1–31 (2023; Zbl 1521.11008], who showed that for t having sufficiently many blocks of 1s in its binary expansion the conjecture holds and the difference s(n+t)−s(n) is approximately normally distributed. In this paper, we are concerned with a related block-counting function r, returning the number of (overlapping) occurrences of the block 11 in the binary expansion of n. Our main result is a similar central limit-type theorem for the difference r(n+t)−r(n), where we give a stronger estimate of the uniform error of Gaussian approximation. As a corollary, we obtain a partial result toward an analogue of Cusick’s conjecture for r: the set of n satisfying r(n+t)≥r(n) has density ≥1/2−ε for t belonging to a set of density 1.