Robert Tichy

TU Graz

Frist Talk

Title:Metric Discrepancy Theory
Abstract:This lecture is related to my graduate course on "Irregularities of Distribution". I will present recent probabilistic results in number theory and in the theory of point distributions. This combines methods from Fourier analysis as well as martingale inequalities and combinatorial techniques. The main results are limit theorems for discrepancies and related counting functions.

Second Talk

Title:Additive Unit Representations
Abstract:In this lecture the following problem is investigated: Let R be a ring (mainly the ring of integers in a number field or function field). When is it possible to represent all r ∈ R as a sum of units: r= u1+... + uk, ui∈ R*. We give sufficient conditions for numbers fields of low order and present a general negative result of Jarden and Narkiewicz. There are also many open problems which are discussed. The proofs depend on tools from algebraic number theory and on diophantine analysis, in particular, on the solution of S-unit equations. Furthermore we establish asymptotic formulas for the number of such representations.