Metric number theory, lacunary series and systems of dilated functions

By a classical result of Weyl, for any increasing sequence $(n_k)_{k \geq 1}$ of integers the sequence of fractional parts $(\{n_k x\})_{k \geq 1}$ is uniformly distributed modulo 1 for almost all $x \in [0,1]$. Except for a few special cases, e.g. when $n_k=k, k \geq 1$, the exceptional set cannot be described explicitly. The exact asymptotic order of the discrepancy of $(\{n_k x\})_{k \geq 1}$ is only known in a few special cases, for example when $(n_k)_{k \geq 1}$ is a (Hadamard) lacunary sequence, that is when $n_{k+1}/n_k \geq q > 1, k \geq 1$. In this case of quickly increasing $(n_k)_{k \geq 1}$ the system $(\{n_k x\})_{k \geq 1}$ (or, more general, $(f(n_k x))_{k \geq 1}$ for a 1-periodic function $f$) shows many asymptotic properties which are typical for the behavior of systems of \emph{independent} random variables. Precise results depend on a fascinating interplay between analytic, probabilistic and number-theoretic phenomena. Without any growth conditions on $(n_k)_{k \geq 1}$ the situation becomes much more complicated, and the system $(f(n_k x))_{k \geq 1}$ will typically fail to satisfy probabilistic limit theorems. An important problem which remains is to study the almost everywhere convergence of series $\sum_{k=1}^\infty c_k f(k x)$, which is closely related to finding upper bounds for maximal $L^2$-norms of the form $$\int_0^1 (\max_{1 \leq M \leq N}| \sum_{k=1}^M c_k f(kx)|^2 dx.$$ The most striking example of this connection is the equivalence of the Carleson convergence theorem and the Carleson--Hunt inequality for maximal partial sums of Fourier series. For general functions $f$ this is a very difficult problem, which is related to finding upper bounds for certain sums involving greatest common divisors.Comment: Survey paper for the RICAM workshop on "Uniform Distribution and Quasi-Monte Carlo Methods", held from October 14-18, 2013, in Linz, Austria. This article will appear in the proceedings volume for this workshop, published as part of the "Radon Series on Computational and Applied Mathematics" by DeGruyte

The density of sets containing large similar copies of finite sets

Funding: VK is supported by the Croatian Science Foundation, project n◦ UIP-2017-05-4129 (MUNHANAP). AY is supported by the Swiss National Science Foundation, grant n◦ P2SKP2 184047.We prove that if E⊆Rd (d≥2) is a Lebesgue-measurable set with density larger than n−2n−1, then E contains similar copies of every n-point set P at all sufficiently large scales. Moreover, 'sufficiently large' can be taken to be uniform over all P with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of n-point sets tends to 1 at a rate 1−O(n−1/5log n).PreprintPeer reviewe

From incommensurate bilayer heterostructures to Allen-Cahn: An exact thermodynamic limit

Assuming any site-potential dependent on two-point correlations, we rigorously derive a new model for an interlayer potential for incommensurate bilayer heterostructures such as twisted bilayer graphene. We use the ergodic property of the local configuration in incommensurate bilayer heterostructures to prove convergence of an atomistic model to its thermodynamic limit without a rate for minimal conditions on the lattice displacements. We provide an explicit error control with a rate of convergence for sufficiently smooth lattice displacements. For that, we introduce the notion of Diophantine 2D rotations, a two-dimensional analogue of Diophantine numbers, and give a quantitative ergodic theorem for Diophantine 2D rotations.Comment: 55 pages, 1 figur