2024/12/20 by Bartosz Sobolewski, Sobolewski, Bartosz
Computer Science · Decision Sciences · #Neural Networks and Applications #Advanced Statistical Process Monitoring
paper · pdf · doi:10.48550/arxiv.2412.15851
Let s(n) denote the sum of binary digits of an integer n ≥ 0. In the recent years there has been interest in the behavior of the differences s(n+t)-s(n), where t ≥ 0 is an integer. In particular, Spiegelhofer and Wallner showed that for t whose binary expansion contains sufficiently many blocks of \mathtt1s the inequality s(n+t) -s(n) ≥ 0 holds for n belonging to a set of asymptotic density >1/2, partially answering a question by Cusick. Furthermore, for such t the values s(n+t) - s(n) are approximately normally distributed. In this paper we consider a natural generalization to the family of block-counting functions Nw, giving the number of occurrences of a block of binary digits w in the binary expansion. Our main result show that for any w of length at least 2 the distribution of the differences Nw(n+t) - Nw(n) is close to a Gaussian when t contains many blocks of \mathtt1s in its binary expansion. This extends an earlier result by the author and Spiegelhofer for w=\mathtt11.