We recall some classical results for uniform distribution modulo one, and relate them with their counterparts in the "localized" setting of correlation functions and gap statistics. We discuss the difficulties arising from the localized setting, with a particular emphasis on questions concerning the almost everywhere behavior of parametric sequences. It turns out that in this metric setting one is naturally led to a Diophantine counting problem, which has interesting connections to additive combinatorics and to moment bounds for the Riemann zeta function.[-]

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.[-]

In the 40s R. Feynman invented a simple model of electron motion, which is now known as Feynman's checkers. This model is also known as the one-dimensional quantum walk or the imaginary temperature Ising model. In Feynman's checkers, a checker moves on a checkerboard by simple rules, and the result describes the quantum-mechanical behavior of an electron.
We solve mathematically a problem by R. Feynman from 1965, which was to prove that the model reproduces the usual quantum-mechanical free-particle kernel for large time, small average velocity, and small lattice step. We compute the small-lattice-step and the large-time limits, justifying heuristic derivations by J. Narlikar from 1972 and by A.Ambainis et al. from 2001. The main tools are the Fourier transform and the stationary phase method.
A more detailed description of the model can be found in Skopenkov M.& Ustinov A. Feynman checkers: towards algorithmic quantum theory. (2020) https://arxiv.org/abs/2007.12879[-]

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.[-]

Exponential Diophantine equations, say of the form (1) $u_{1}+...+u_{k}=b$ where the $u_{i}$ are exponential terms with fixed integer bases and unknown exponents and b is a fixed integer, play a central role in the theory of Diophantine equations, with several applications of many types. However, we can bound the solutions only in case of k = 2 (by results of Gyory and others, based upon Baker's method), for k > 2 only the number of so-called non-degenerate solutions can be bounded (by the Thue-Siegel-Roth-Schmidt method; see also results of Evertse and others). In particular, there is a big need for a method which is capable to solve (1) completely in concrete cases.
Skolem's conjecture (roughly) says that if (1) has no solutions, then it has no solutions modulo m with some m. In the talk we present a new method which relies on the principle behind the conjecture, and which (at least in principle) is capable to solve equations of type (1), for any value of k. We give several applications, as well. Then we provide results towards the solution of Skolem's conjecture. First we show that in certain sense it is 'almost always' valid. Then we provide a proof for the conjecture in some cases with k = 2, 3. (The handled cases include Catalan's equation and Fermat's equation, too - the precise connection will be explained in the talk). Note that previously Skolem's conjecture was proved only for k = 1, by Schinzel.
The new results presented are (partly) joint with Bertok, Berczes, Luca, Tijdeman.[-]