Block occurrences in the binary expansion

The binary sum-of-digits function [Formula: see text] returns the number of ones in the binary expansion of a nonnegative integer. Cusick’s Hamming weight conjecture states that for all integers [Formula: see text] the set of integers [Formula: see text] such that [Formula: see text] has asymptotic density strictly larger than [Formula: see text]. So far, the strongest results are due to Spiegelhofer and Wallner, who showed that for t having sufficiently many blocks of [Formula: see text]s in its binary expansion the conjecture holds and the difference [Formula: see text] is approximately normally distributed. In this paper, we are concerned with a related block-counting function [Formula: see text], returning the number of (overlapping) occurrences of the block [Formula: see text] in the binary expansion of n. Our main result is a similar central limit-type theorem for the difference [Formula: see text], 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 [Formula: see text]: the set of n satisfying [Formula: see text] has density [Formula: see text] for t belonging to a set of density 1.