Normality, Computability and Randomness

Robert F. Tichy
Technische Universität Graz

Date(s) : 14/04/2015   iCal
11 h 00 min - 11 h 40 min

In the first part we start from the concept of normal numbers with respect to a given base and present some recent methods of explicit constructions. Furthermore, we present a dynamical interpretation and extensions to absolutely normal numbers. This concept is investigated from a computational point of view, extending some results of A. Turing. The second part of the lecture is devoted to van der Corput sets and sets of recurrence. We give new constructions involving equidistribution of sequences of prime powers. This refines and unifies earlier results obtained by Sárközy, Furstenberg, Kamae and Mendés-France and Bergelson and Lesigne. The proofs heavily depend on analytic machinery involving bounds for exponential sums. In the third part of the lecture we give some new sharp result concerning the probabilistic behaviour of lacunary sequences. In particular we show limit theorems for discrepancy functions and related quantities. This leads to quantitative results in the theory of normal numbers and pseudorandomness.



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