Related papers: Higher dimensional Steinhaus and Slater problems v…

We provide conjectural necessary and (separately) sufficient conditions for the Hilbert scheme of points of a given length to have the maximum dimension tangent space at a point. The sufficient condition is claimed for 3D and reduces the…

This paper is motivated by two problems in the theory of Diophantine approximation, namely, Davenport's problem regarding badly approximable points on submanifolds of a Euclidean space and Schmidt's problem regarding the intersections of…

We establish the Strauss conjecture for nontrapping obstacles when the spatial dimension $n$ is two. As pointed out in \cite{HMSSZ} this case is more subtle than $n=3$ or 4 due to the fact that the arguments of the first two authors…

We give a short proof of the fact that there are no measurable subsets of Euclidean space (in dimension d > 2), which, no matter how translated and rotated, always contain exactly one integer lattice point. In dimension d=2 (the original…

A classical theorem due to Mattila (see \cite{Mat84}; see also \cite{M95}, Chapter 13) says that if $A,B \subset {\Bbb R}^d$ of Hausdorff dimension $s_A, s_B$, respectively, with $s_A+s_B \ge d$, $s_B>\frac{d+1}{2}$ and $dim_{{\mathcal…

In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable…

We consider the problem of digitalizing Euclidean segments. Specifically, we look for a constructive method to connect any two points in $\mathbb{Z}^d$. The construction must be {\em consistent} (that is, satisfy the natural extension of…

We explain the triangular gaps observed experimentally in the most popular sizes of the $h$-fold iterated sumset, $hA,$ when $A$ is a randomly chosen four-element subset of the first $q$ natural numbers, for $q$ much larger than $h.$

This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear subspaces generated by the best…

The fundamental gap conjecture proved by Andrews and Clutterbuck in 2011 provides the sharp lower bound for the difference between the first two Dirichlet Laplacian eigenvalues in terms of the diameter of a convex set in $\mathbb{R}^N$. The…

We consider overdamped Langevin diffusions in Euclidean space, with curvature equal to the spectral gap. This includes the Ornstein-Uhlenbeck process as well as non-Gaussian and non-product extensions with convex interaction, such as the…

For $s=3,4$, we prove the existence of arbitrarily long sequences of consecutive integers none of which is a sum of $s$ nonnegative $s$-th powers. More generally, we study the existence of gaps between the values $\leq N$ of diagonal forms…

We explore a comprehensive analysis of the formalism governing the gravitational field equations in degenerate higher-order scalar-tensor theories. The propagation of these theories in the vacuum has a maximum of three degrees of freedom…

In this paper we prove an upper bound on the "size" of the set of multiplicatively $\psi$-approximable points in $\mathbb R^d$ for $d>1$ in terms of $f$-dimensional Hausdorff measure. This upper bound exactly complements the known lower…

The study of Coulomb branches of 3-dimensional N=4 gauge theories via the associated Hilbert series, the so-called monopole formula, has been proven useful not only for 3-dimensional theories, but also for Higgs branches of 5 and…

In this paper we investigate the Erd\"os/Falconer distance conjecture for a natural class of sets statistically, though not necessarily arithmetically, similar to a lattice. We prove a good upper bound for spherical means that have been…

We locate gaps in the spectrum of a Hamiltonian on a periodic cuboidal (and generally hyperrectangular) lattice graph with $\delta$ couplings in the vertices. We formulate sufficient conditions under which the number of gaps is finite. As…

The 3SUM problem represents a class of problems conjectured to require $\Omega (n^2)$ time to solve, where $n$ is the size of the input. Given two polygons $P$ and $Q$ in the plane, we show that some variants of the decision problem,…

Our goal is to show that both the fast and slow versions of the triangle map (a type of multi-dimensional continued fraction algorithm) in dimension $n$ are ergodic, resolving a conjecture of Messaoudi, Noguiera and Schweiger. This…

For $\lambda \in (1/2, 1)$ and $\alpha$, we consider sets of numbers $x$ such that for infinitely many $n$, $x$ is $2^{-\alpha n}$-close to some $\sum_{i=1}^n \omega_i \lambda^i$, where $\omega_i \in \{0,1\}$. These sets are in Falconer's…