(Logarithmic) densities for automatic sequences along primes and squares

TL;DR

This paper introduces a method to transfer density results from primitive automatic sequences to general automatic sequences along primes and squares, establishing the existence and computability of their logarithmic densities, with specific results for prime densities.

Contribution

It develops a new transfer method for density results and provides criteria for the existence and computability of densities along primes and squares for automatic sequences.

Findings

Logarithmic densities along squares and primes exist and are computable.

Prime densities are always rational.

Automatic sequences are orthogonal to bounded multiplicative aperiodic functions.

Abstract

In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic sequence along squares $(n^2)_{n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable. Furthermore, we give for these subsequences a criterion to decide whether the densities exist, in which case they are also computable. In particular in the prime case these densities are all rational. We also deduce from a recent result of the third author and Lema\'nczyk that all subshifts generated by automatic sequences are orthogonal to any bounded multiplicative aperiodic function.