Given a planar domain on the rectangular grid, how many ways are there of tiling it by dominos (that is, by 1x2 rectangles)? And how does a generic tiling of a given domain look like?

It turns out that these questions are related to the determinants-based formulas, and that likewise formulas appear in many similar situations. In this direction, one obtains the famous arctic circle theorem, describing the behaviour of a generic domino tiling of an aztec diamond, and a statement for the lozenges tilings on the hexagonal lattice, giving the shape of a corner of a cubic crystal.

In an award winning 2014 paper, Damanik, Fillman, and Gorodetski rigorously established a framework for investigating Schrodinger operators on the real line whose potentials are generated by ergodic subshifts. In the case of the Fibonacci subshift, they also described the asymptotic behavior in the large energy and small coupling settings when the potential pieces are characteristic functions of intervals of equal length. These estimates relied on explicit formulae and calculations, and thus could not be immediately generalized. In joint work with Fillman, we show that when the potential pieces are square integrable, the Hausdorff dimension of the spectrum tends to one in the large energy and small coupling settings.

Recently there has been a remarkable progress in understanding projections of many concrete fractals sets and measures. In this talk we will discuss some of these results and techniques, and also some related open problems.

Dirichlet’s approximation theorem states that for every real number x there exist infinitely many rationals p/q with |x-p/q| < 1/q^2. If x is in the unit interval, then viewing rationals as algebraic numbers of degree 1, q is also the height of its primitive integer polynomial, where height means the maximum of the absolute values of the coefficients. This suggests a more general question: How well can real numbers be approximated by algebraic numbers of degree at most n, relative to their heights? We will discuss Wirsing’s conjecture which proposes an answer to this question and Schmidt and Davenport’s proof of the n = 2 case, as well as some open questions.

We show that for any two homogeneous Cantor sets with sum of Hausdorff dimensions that exceeds 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of homogeneous Cantor sets). We will also discuss the connection of this problem with the question on properties of one dimensional self-similar sets with overlaps.