Subsequences of automatic sequences indexed by⌊nc⌋and correlations

Abstract

The main goal of this paper is to study the behavior of subsequences u c = { u ( ⌊ n c ⌋ ) : n ∈ N } of automatic sequences u that are indexed by ⌊ n c ⌋ for some c > 1 . In particular we show that the densities of the letters of u c are precisely the same as those of the original sequence (provided that c < 7 / 5 ). In this sense u c and u behave in the same way. However, the pair correlation might be completely different as we will show in the special case of the Thue–Morse sequence. The proofs use exponential sum estimates like the double large sieve and a discrete Fourier analysis related to automatic sequences.