Let $\alpha$ $\epsilon$ $\mathbb{R}^d$ be a vector whose entries $\alpha_1, . . . , \alpha_d$ and $1$ are linearly independent over the rationals. We say that $S \subset \mathbb{T}^d$ is a bounded remainder set for the sequence of irrational rotations $\lbrace n\alpha\rbrace_{n\geqslant1}$ if the discrepancy
$ \sum_{k=1}^{N}1_S (\lbrace k\alpha\rbrace) - N$ $mes(S)$
is bounded in absolute value as $N \to \infty$. In one dimension, Hecke, Ostrowski and Kesten characterized the intervals with this property.
We will discuss the bounded remainder property for sets in higher dimensions. In particular, we will see that parallelotopes spanned by vectors in $\mathbb{Z}\alpha + \mathbb{Z}^d$ have bounded remainder. Moreover, we show that this condition can be established by exploiting a connection between irrational rotation on $\mathbb{T}^d$ and certain cut-and-project sets. If time allows, we will discuss bounded remainder sets for the continuous irrational rotation $\lbrace t \alpha : t$ $\epsilon$ $\mathbb{R}^+\rbrace$ in two dimensions.[-]

Bounded remainder sets for a dynamical system are sets for which the Birkhoff averages of return times differ from the expected values by at most a constant amount. These sets are rare and important objects which have been studied for over 100 years. In the last few years there have been a number of results which culminated in explicit constructions of bounded remainder sets for toral rotations in any dimension, of all possible allowable volumes. In this talk we are going to explain these results, and then explain how to generalize them to give explicit constructions of bounded remainder sets for rotations in $p$-adic solenoids. Our method of proof will make use of a natural dynamical encoding of patterns in non-Archimedean cut and project sets.[-]

Discrepancy is a measure of equidistribution for sequences of points. We consider here discrepancy in the setting of symbolic dynamics and we discuss the existence of bounded remainder sets for some families of zero entropy subshifts, from a topological dynamics viewpoint. A bounded remainder set is a set which yields bounded discrepancy, that is, the number of times it is visited differs by the expected time only by a constant. Bounded discrepancy provides particularly strong convergence properties of ergodic sums. It is also closely related to the notions of balance in word combinatorics.[-]

Discrepancy and discrete energy are two of the most standard ways to measure the quality of the distribution of a finite point set, and it is very well known that there is strong interplay between these concepts. One particular important example of such interplay is the classical Stolarsky principle which ties together the pairwise sum of distances and the spherical cap discrepancy. In the current talk we shall survey various manifestations of this connection between discrepancy and energy minimization.[-]

We are primarily concerned with the distribution of billiard orbits and geodesics in nonintegrable flat dynamical systems, those that exhibit split singularities. This is an area which has been studied over a number of decades by different authors, and many deep results have been obtained.We are particularly interested in results concerning density and uniformity. However, until very recently, the major known results are time-qualitative and do not seem to say anything definite about the necessary time range. Their proofs all use Birkhoff's ergodic theorem, which is quantitative in the sense that it can tell us precisely how often some relevant event is going to take place, but time-qualitative in the sense that it requires unlimited time range.This leads to a very natural question as to what can happen in a realistic finite time scale.In these two talks, we describe a new non-ergodic method which is a combination of combinatorics, number theory and linear algebra, and which leads to time-quantitative results concerning density and uniformity of some billiard orbits and geodesics in many nonintegrable flat dynamical systems. We discuss two versions of the technique, an eigenvaluebased version as well as an eigenvalue-free version which relies on size magnification.This is work with Jozsef Beck, Michael Donders and Yuxuan Yang.[-]

In this talk I will report on recent progress on two different problems in discrepancy theory. In the first part I will present a recent extension of the notion of jittered sampling to arbitrary partitions of the unit cube. In this joint work with Markus Kiderlen from Aarhus, we introduce the notion of a uniformly distributed triangular array. Moreover, we show that the expected Lp-discrepancy of a point sample generated from an arbitrary equi volume partition of the unit cube is always strictly smaller than the expected Lp-discrepancy of a set of N uniform random samples for p > 1.
The second part of the talk is dedicated to greedy energy minimization. I will give a new characterisation of the classical van der Corput sequence in terms of a minimization problem and will discuss various related open questions.[-]

We are primarily concerned with the distribution of billiard orbits and geodesics in nonintegrable flat dynamical systems, those that exhibit split singularities. This is an area which has been studied over a number of decades by different authors, and many deep results have been obtained.We are particularly interested in results concerning density and uniformity. However, until very recently, the major known results are time-qualitative and do not seem to say anything definite about the necessary time range. Their proofs all use Birkhoff's ergodic theorem, which is quantitative in the sense that it can tell us precisely how often some relevant event is going to take place, but time-qualitative in the sense that it requires unlimited time range.This leads to a very natural question as to what can happen in a realistic finite time scale.In these two talks, we describe a new non-ergodic method which is a combination of combinatorics, number theory and linear algebra, and which leads to time-quantitative results concerning density and uniformity of some billiard orbits and geodesics in many nonintegrable flat dynamical systems. We discuss two versions of the technique, an eigenvaluebased version as well as an eigenvalue-free version which relies on size magnification.This is work with Jozsef Beck, Michael Donders and Yuxuan Yang.[-]

In this talk I will report on recent progress on two different problems in discrepancy theory. In the first part I will present a recent extension of the notion of jittered sampling to arbitrary partitions of the unit cube. In this joint work with Markus Kiderlen from Aarhus, we introduce the notion of a uniformly distributed triangular array. Moreover, we show that the expected Lp-discrepancy of a point sample generated from an arbitrary equi volume partition of the unit cube is always strictly smaller than the expected Lp-discrepancy of a set of N uniform random samples for p > 1.
The second part of the talk is dedicated to greedy energy minimization. I will give a new characterisation of the classical van der Corput sequence in terms of a minimization problem and will discuss various related open questions.[-]

In this talk I will report on recent progress on two different problems in discrepancy theory.In the first part I will present a recent extension of the notion of jittered sampling to arbitrary partitions of the unit cube. In this joint work with Markus Kiderlen from Aarhus, we introduce the notion of a uniformly distributed triangular array. Moreover, we show that the expected Lp-discrepancy of a point sample generated from an arbitrary equi volume partition of the unit cube is always strictly smaller than the expected Lp-discrepancy of a set of N uniform random samples for p > 1.
The second part of the talk is dedicated to greedy energy minimization. I will give a new characterisation of the classical van der Corput sequence in terms of a minimization problem and will discuss various related open questions.[-]

This is a survey on progress in metric discrepancy theory and probabilistic aspects in harmonic analysis. We start with classical limit theorems of Salem and Zygmund as well as with the work of Erdoes and Gaal and of Walter Philipp. A focus lies on laws of the iterated logarithm for discrepancy functions of lacunary sequences. We show the connection to certain diophantine properties of the underlying lacunary sequences obtaining precise asymptotic formulas. Different phenomena for subexponentially growing, for exponentially growing and for superexponentially growing sequences are established. Furthermore, relations to arithmetic dynamical systems and to Donald Knuth`s concept of pseudorandomness are discussed. Recent results are contained in joint work with Christoph Aistleitner and Istvan Berkes and it is planed to publish parts of it in a Jean Morlet Springer lecture Notes volume.[-]