Subsequences of automatic sequences and uniform distribution

Abstract

Abstract. Automatic sequences and their number theoretic properties have been intensively studied during the last 20 or 30 years. Since automatic sequences are quite regular (they just have linear subword complexity) they cannot be used as quasi-random sequences. However, the situation changes drastically when one uses proper subsequences, for example the subsequence along primes or squares. It is conjectured that the resulting sequences are normal sequences which could be already proved for the Thue-Morse sequence along the subsequence of squares.This kind of research is very challenging and was mainly motivated by the Gelfond problems for the sum-of-digits function. In particular during the last few years there was a spectacular progress due to the Fourier analytic method by Mauduit and Rivat. In this article we survey these recent developments, comment on the proof methods and formulate quite general conjectures. We also present a new result on the subsequence along primes of so-called invertible automatic sequences.

On the pseudorandomness of automatic sequences

“…For more details on the normality of automatic sequences and their subsequences we refer to [31].…”

“…These sums can be estimated via a Fourier analytic method of Mauduit and Rivat which has its origin in [58,59]. For more details we refer to the survey [31] of Drmota and the original papers [32,71].…”

“…A k-automatic sequence (s n ) over an alphabet Σ is the output sequence of a finite automaton, where the input is the k-ary digital expansion of n. Automatic sequences have gained much attention during the last decades. For monographs and surveys about automatic sequences we refer to [1,2,7,8].…”

Möbius orthogonality for the Zeckendorf sum-of-digits function

“…This conjecture has received a lot of attention during the last years and could be proved for several instances [3,4,5,9,11,16,17,18,20,21,22,23,24,26,29,31,35,38,41]. In particular special automatic sequences were handled recently: [8,10,12,15,19,25,27,32,33]. And finally the second author could solve the Sarnak conjecture for all automatic sequences [34].…”

On the Nth linear complexity of automatic sequences

“…A k-automatic sequence (u n ) over an alphabet A is the output sequence of a finite automaton, where the input is the k-ary digital expansion of n. Automatic sequences have gained much attention during the last decades. For monographs and surveys about automatic sequences we refer to [1,2,12,13].…”