Related papers: Ramsey numbers and extremal structures in polar sp…

Let $X$ be a complete variety of dimension $n$ over an algebraically closed field $\mathbf{K}$. Let $V_\bullet$ be a graded linear series associated to a line bundle $L$ on $X$, that is, a collection $\{V_m\}_{m\in\mathbb{N}}$ of vector…

We generalize ultraproducts and local-global limits of graphs to hypergraphs and other structures. We show that the local statistics of an ultraproduct of a sequence of hypergraphs are the ultralimits of the local statistics of the…

We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered: (1) we improve the Ray-Chaudhuri--Wilson bound of the size of uniform intersecting families of…

This paper presents three new families of fractional Sobolev spaces and their accompanying theory in one-dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural…

We consider an elliptic boundary problem over a bounded region $\Omega$ in $\mathbb{R}^n$ and acting on the generalized Sobolev space $W^{0,\chi}_p(\Omega)$ for $1 < p < \infty$. We note that similar problems for $\Omega$ either a bounded…

Let $\mathbb X$ and $\mathbb Y$ be $\ell$-connected Jordan domains, $\ell \in \mathbb N$, with rectifiable boundaries in the complex plane. We prove that any boundary homeomorphism $\varphi \colon \partial \mathbb X \to \partial \mathbb Y$…

A nearly logarithmic lower bound on the randomized competitive ratio for the metrical task systems problem is presented. This implies a similar lower bound for the extensively studied k-server problem. The proof is based on Ramsey-type…

We develop the theory of tamed spaces which are Dirichlet spaces with distribution-valued lower bounds on the Ricci curvature and investigate these from an Eulerian point of view. To this end we analyze in detail singular perturbations of…

In this paper we prove the existence of extremal functions for the Adams-Moser-Trudinger inequality on the Sobolev space $H^{m}(\Omega)$, where $\Omega$ is any bounded, smooth, open subset of $\mathbb{R}^{2m}$, $m\ge 1$. Moreover, we extend…

We give a precise estimate for the number of lattice points in certain bounded subsets of $\mathbb{R}^{n}$ that involve `hyperbolic spikes' and occur naturally in multiplicative Diophantine approximation. We use Wilkie's o-minimal structure…

We study permutations in $S_n$ that simultaneously avoid the pattern $132$ and satisfy the adjacency bound $|\pi_{i+1} - \pi_i| \leq m$ for all $i$, denoting their number by $A_n^{(m)}$. This combination of a global pattern restriction and…

We study the bias of random bounded-degree polynomials over odd prime fields and show that, with probability exponentially close to 1, such polynomials have exponentially small bias. This also yields an exponential tail bound on the weight…

A remarkable coincidence has led to the discovery of a family of packings of (m^2+m-2) m/2-dimensional subspaces of m-dimensional space, whenever m is a power of 2. These packings meet the ``orthoplex bound'' and are therefore optimal.

We elaborate on the intimate connection between the largest volume of an empty axis-parallel box in a set of $n$ points from $[0,1]^d$ and cover-free families from the extremal set theory. This connection was discovered in a recent paper of…

We apply ideas from the theory of limits of dense combinatorial structures to study order types, which are combinatorial encodings of finite point sets. Using flag algebras we obtain new numerical results on the Erd\H{o}s problem of finding…

We study elliptic equations of order $2m$ with nonlocal boundary-value conditions in plane angles and in bounded domains, dealing with the case where the support of nonlocal terms intersects the boundary. We establish necessary and…

We improve the best known upper bound for the bracketing number of $d$-dimensional axis-parallel boxes anchored in $0$ (or, put differently, of lower left orthants intersected with the $d$-dimensional unit cube $[0,1]^d$). More precisely,…

It is known that the lengths of closed geodesics of an arithmetic hyperbolic orbifold are related to Salem numbers. We initiate a quantitative study of this phenomenon. We show that any non-compact arithmetic $3$-dimensional orbifold…

We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o-minimal structure. This fact together with the results in a previous paper implies tame dimension theory and…

We construct a family of non-parametric (infinite-dimensional) manifolds of finite measures on $R^d$. The manifolds are modelled on a variety of weighted Sobolev spaces, including Hilbert-Sobolev spaces and mixed-norm spaces. Each supports…