Automatic sequences, generalised polynomials, and nilmanifolds
摘要：Generalised polynomials are expressions formed from ordinary polynomials using the integer part operation, addition and multiplication. Their distribution has been extensively studied, and thanks to the work of Bergelson and Leibman is known to be intimately connected to dynamics on nilmanifolds. Automatic sequences are those whose n-th term can be computed by a finite machine given the digits of n on input. The starting point for the project whose results I will discuss is the following simple-looking question: Are there any non-trivial examples of automatic sequences given by generalised polynomial formulas? I will present a partial solution to this, as well as examples of sequences which somewhat surprisingly turn out to be generalised polynomials.