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 $n$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.