Subsequences of automatic sequences indexed by ?n ~c? and correlations

摘要

The main goal of this paper is to study the behavior of subsequences uc={u(?nc?):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.