Related papers: A note on the no-$(d+2)$-on-a-sphere problem

Two-dimensional $O(N)$ non-linear sigma models are exactly solvable theories and have many applications, from statistical mechanics to their use as QCD toy models. We consider a supersymmetric extension, the non-linear sigma model on the…

We show that among antipodal $2d$-point configurations on the sphere $S^{d-1}$ in $\mathbb R^d$, the set of vertices of a regular cross-polytope inscribed in $S^{d-1}$ uniquely solves the best-covering problem (this is new for $d\geq 5$)…

A well known consequence of the Borsuk-Ulam theorem is that if the $d$-dimensional sphere $S^d$ is covered with less than $d+2$ open sets, then there is a set containing a pair of antipodal points. In this paper we provide lower and upper…

We improve the exponent for the discrete Fourier restriction to the $n$ dimensional sphere, from $p=\frac{2(n+1)}{n-3}$ to $p=\frac{2n}{n-3}$, when $n\ge 4$.

We prove that any set of points in $\mathbb{R}^d$, any three of which form an angle less than $\frac{\pi}{3} + c$, has size $(1+\Theta(c))^d$ for sufficiently small $c>0$. The proof is based on a refinement of an approach by Erd\H{o}s and…

If $\mathcal{B}\subset \mathbb{R}^d$ ($d\geqslant 2$) is a compact convex domain with a smooth boundary of finite type, we prove that for almost every rotation $\theta\in SO(d)$ the remainder of the lattice point problem, $P_{\theta…

For $d\ge 1$, we construct a compact subset $K\subseteq \mathbb {R}^{d+1}$ containing a $d$-sphere of every radius between $1$ and $2$, such that for every $\delta\in (0,1)$, the $\delta$-neighbourhood of $K$ has Lebesgue measure $\lesssim…

In this paper we present three different results dealing with the number of $(\leq k)$-facets of a set of points: 1. We give structural properties of sets in the plane that achieve the optimal lower bound $3\binom{k+2}{2}$ of $(\leq…

We consider the space $[0,n]^3$, imagined as a three dimensional, axis-aligned grid world partitioned into $n^3$ $1\times 1 \times 1$ unit cubes. Each cube is either considered to be empty, in which case a line of sight can pass through it,…

Let $(P,E)$ be a $(d+1)$-uniform geometric hypergraph, where $P$ is an $n$-point set in general position in $\mathbb{R}^d$ and $E\subseteq {P\choose d+1}$ is a collection of $\epsilon{n\choose d+1}$ $d$-dimensional simplices with vertices…

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…

We consider the equation $d^2\Delta u - u+ u^{\frac{n-k+2}{n-k-2}} =0\,\hbox{in}\Omega $, under zero Neumann boundary conditions, where $\Omega$ is open, smooth and bounded and $d$ is a small positive parameter. We assume that there is a…

We will show that for any $n\ge N$ points on the $N$-dimensional sphere $S^N$ there is a closed hemisphere which contains at least $\lfloor\frac{n+N+1}{2}\rfloor$ of these points. This bound is sharp and we will calculate the amount of sets…

In 1995, Jockusch constructed an infinite family of centrally symmetric $3$-dimensional simplicial spheres that are cs-$2$-neighborly. Here we generalize his construction and show that for all $d\geq 3$ and $n\geq d+1$, there exists a…

Spherical t-designs are Chebyshev-type averaging sets on the d-sphere S^d which are exact for polynomials of degree at most t. This concept was introduced in 1977 by Delsarte, Goethals, and Seidel, who also found the minimum possible size…

Eberhard and Pohoata conjectured that every $3$-cube-free subset of $[N]$ has size less than $2N/3+o(N)$. In this paper we show that if we replace $[N]$ with $\mathbb{Z}_N$ the upper bound of $2N/3$ holds, and the bound is tight when $N$ is…

We show that there exists a lattice covering of $\mathbb{R}^n$ by Eucledian spheres of equal radius with density $O\big(n \ln^{\beta} n \big)$ as $n\to\infty$, where \begin{align*} \beta := \frac{1}{2} \log_2 \left(\frac{8 \pi…

By definition, a rigid graph in $\mathbb{R}^d$ (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system.…

In 2009, Roeglin and Teng showed that the smoothed number of Pareto optimal solutions of linear multi-criteria optimization problems is polynomially bounded in the number $n$ of variables and the maximum density $\phi$ of the semi-random…

In 2008, Bukh, Matousek, and Nivasch conjectured that for every n-point set S in R^d and every k, 0 <= k <= d-1, there exists a k-flat f in R^d (a "centerflat") that lies at "depth" (k+1) n / (k+d+1) - O(1) in S, in the sense that every…