Roots of unity of order dividing $n$ equidistribute around the unit circle as $n$ tends to infinity. With some extraeffort the same can be shown when restricting to roots of unity of exact order $n$. Equidistribution is measured by comparing the average of a continuous test function evaluated at these roots of unity with the integral over the complex unit circle. Baker, Ih, and Rumely extended this to test function with logarithmic singularities of the form $\log|P|$ where $P$ is a univariate polynomial in algebraic coefficients. I will discuss joint work with Vesselin Dimitrov where we allow $P$ to come from a class of a multivariate polynomials, extending a result of Lind, Schmidt, and Verbitskiy. Our method draws from earlier work of Duke.[-]

As is well known, simultaneous rational approximations to the values of smooth functions of real variables involve counting and/or understanding the distribution of rational points lying near the manifold parameterised by these functions. I will discuss recent results in this area regarding lower bounds for the Hausdorff dimension of $\tau$-approximable values, where $\tau\geq \geq 1/n$ is the exponent of approximations. In particular, I will describe a very recent development for non-degenerate maps as well as a recently introduced simple technique based on the so-called Mass Transference Principle that surprisingly requires no conditions on the functions except them being $C^2$.[-]

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.[-]