Related papers: On the lower bounds for the spherical cap discrepa…

For each $N\ge C_dt^d$ we prove the existence of a well separated spherical $t$-design in the sphere $S^d$ consisting of $N$ points, where $C_d$ is a constant depending only on $d$.

A spherical two-distance set is a finite collection of unit vectors in $\reals^n$ such that the set of distances between any two distinct vectors has cardinality two. We use the semidefinite programming method to compute improved estimates…

The Tomas-Stein inequality for a compact subset $\Gamma$ of the sphere $S^d$ states that the mapping $f\mapsto \widehat{f\sigma}$ is bounded from $L^2(\Gamma,\sigma)$ to $L^{2+4/d}(\R^{d+1})$. Then conditional on a strict comparison between…

Let $S \subset {\mathbb R}^d$ be contained in the unit ball. Let $\Delta(S)=\{||a-b||:a,b \in S\}$, the Euclidean distance set of $S$. Falconer conjectured that the $\Delta(S)$ has positive Lebesque measure if the Hausdorff dimension of $S$…

We find sharp absolute constants $C_1$ and $C_2$ with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval…

A new lower bound on the average reconstruction error variance of multidimensional sampling and reconstruction is presented. It applies to sampling on arbitrary lattices in arbitrary dimensions, assuming a stochastic process with constant,…

We study the lower bound for Koldobsky's slicing inequality. We show that there exists a measure $\mu$ and a symmetric convex body $K \subseteq \mathbb{R}^n$, such that for all $\xi\in S^{n-1}$ and all $t\in \mathbb{R},$…

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,…

We prove the existence of an open set $\Omega\subset\mathbb{S}^2$ for which the first positive eigenvalue of the Laplacian with Neumann boundary condition exceeds that of the geodesic disk having the same area. This example holds for large…

We evaluate the variance of the number of lattice points in a small randomly rotated spherical ball on a surface of 3-dimensional sphere centered at the origin. Previously, Bourgain, Rudnick, and Sarnak showed conditionally on the…

Given a sphere of any radius $r$ in an $n$-dimensional Euclidean space, we study the coverings of this sphere with solid spheres of radius one. Our goal is to design a covering of the lowest covering density, which defines the average…

We consider the hypergraph Tur\'an problem of determining $\mathrm{ex}(n, S^d)$, the maximum number of facets in a $d$-dimensional simplicial complex on $n$ vertices that does not contain a simplicial $d$-sphere (a homeomorph of $S^d$) as a…

Mordechay B. Levin has constructed a number $\lambda$ which is normal in base 2, and such that the sequence $(\left\{2^n \lambda\right\})_{n=0,1,2,\ldots}$ has very small discrepancy $D_N$. Indeed we have $N\cdot D_N = \mathcal{O}…

For several weights based on lattice point constructions in $\mathbb{R}^d (d \geq 2)$, we prove that the sharp $L^2$ weighted restriction inequality for the sphere is very different than the corresponding result for the paraboloid. The…

The following conjecture has been known for many decades as Schiffer's symmetry problem (or Schiffer's conjecture): Assume that $\Delta u+k^2u=0$ in $D$, $u|_S=0$, $u_N|_S=1$, where $D\subset \mathbb{R}^3$ is a bounded, connected,…

We study the spherical cap packing problem with a probabilistic approach. Such probabilistic considerations result in an asymptotic sharp universal uniform bound on the maximal inner product between any set of unit vectors and a…

We consider the imbedding inequality || f ||_{L^r(R^d)} <= S_{r,n,d} || f ||_{H^{n}(R^d)}; H^{n}(R^d) is the Sobolev space (or Bessel potential space) of L^2 type and (integer or fractional) order n. We write down upper bounds for the…

We consider capillarity functionals which measure the perimeter of sets contained in a Euclidean half-space assigning a constant weight $\lambda \in (-1,1)$ to the portion of the boundary that touches the boundary of the half-space.…

Let the kissing number $K(d)$ be the maximum number of non-overlapping unit balls in $\mathbb R^d$ that can touch a given unit ball. Determining or estimating the number $K(d)$ has a long history, with the value of $K(3)$ being the subject…

In terms of the minimal $N$-point diameter $D_d(N)$ for $R^d,$ we determine, for a class of continuous real-valued functions $f$ on $[0,+\infty],$ the $N$-point $f$-best-packing constant $\min\{f(\|x-y\|)\, :\, x,y\in \R^d\}$, where the…