On the pseudorandomness of Parry--Bertrand automatic sequences

TL;DR

This paper examines the pseudorandomness of Parry--Bertrand automatic sequences, revealing they have large correlation measures and are not pseudorandom, with differences observed between even- and odd-order measures.

Contribution

It provides the first analysis of correlation measures for a family of morphic sequences, including classical automatic sequences, showing their non-pseudorandom nature.

Findings

Sequences have large even-order correlation measures.

Automatic sequences are far from pseudorandom due to high correlation measures.

Different behaviors of correlation measures observed between even and odd orders.

Abstract

The correlation measure is a testimony of the pseudorandomness of a sequence $\infw{s}$ and provides information about the independence of some parts of $\infw{s}$ and their shifts. Combined with the well-distribution measure, a sequence possesses good pseudorandomness properties if both measures are relatively small. In combinatorics on words, the famous $b$-automatic sequences are quite far from being pseudorandom, as they have small factor complexity on the one hand and large well-distribution and correlation measures on the other. This paper investigates the pseudorandomness of a specific family of morphic sequences, including classical $b$-automatic sequences. In particular, we show that such sequences have large even-order correlation measures; hence, they are not pseudorandom. We also show that even- and odd-order correlation measures behave differently when considering some simple morphic sequences.