Discrete correlations of order 2 of generalised Rudin-Shapiro sequences: a combinatorial approach
Abstract
We introduce a family of block-additive automatic sequences, that are obtained by allocating a weight to each couple of digits, and defining the th term of the sequence as being the total weight of the integer written in base . Under an additional difference condition on the weight function, these sequences can be interpreted as generalised Rudin-Shapiro sequences, and we prove that they have the same correlations of order 2 as sequences of symbols chosen uniformly and independently at random. The speed of convergence is very fast and is independent of the prime factor decomposition of . This extends recent work of Tahay. The proof relies on direct observations about base- representations of integers and combinatorial considerations. We also provide extensions of our results to higher-dimensional block-additive sequences.
Keywords
Cite
@article{arxiv.2006.13162,
title = {Discrete correlations of order 2 of generalised Rudin-Shapiro sequences: a combinatorial approach},
author = {Irène Marcovici and Thomas Stoll and Pierre-Adrien Tahay},
journal= {arXiv preprint arXiv:2006.13162},
year = {2020}
}