Discrete correlations of order 2 of generalised Rudin-Shapiro sequences: a combinatorial approach

TL;DR

This paper introduces a family of block-additive automatic sequences generalizing Rudin-Shapiro sequences, demonstrating they have similar second-order correlations to random sequences with rapid convergence, using combinatorial methods.

Contribution

It extends the understanding of correlations in generalized Rudin-Shapiro sequences and provides a combinatorial proof approach with extensions to higher dimensions.

Findings

Sequences have correlations of order 2 similar to random sequences

Convergence speed is fast and independent of prime factorization of k

Results extend to higher-dimensional sequences

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 $n$th term of the sequence as being the total weight of the integer $n$ written in base $k$. 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 $k$. This extends recent work of Tahay. The proof relies on direct observations about base-$k$ representations of integers and combinatorial considerations. We also provide extensions of our results to higher-dimensional block-additive sequences.