Bence Borda: Uniform distribution in discrete and continuous time

The simplest dynamical systems on the d-dimensional torus are the translation (discrete time) and the linear flow (continuous time). We survey some old and some recent results concerning quantitative forms of uniform distribution of the orbits of these systems. Uniformity with respect to polytopes, convex sets and Lebesgue measurable sets turn out to be very different in discrete and in continuous time. In particular, a recent result of the speaker shows that there is no "irregularities of distribution" for continuous curves on the torus in any dimension.