Related papers: The density of planar sets avoiding unit distances

It has been conjectured, on the basis of the gauge-gravity duality, that the ratio of the shear viscosity to the entropy density should be universally bounded from below by 1/ 4 pi in units of the Planck constant divided by the Boltzmann…

We study density thresholds that force a measurable set $E\subseteq\mathbb{R}^d$ to contain all sufficiently large similar copies of every $n$-point configuration. We prove a lower bound of the form $1-O((\log n)/n)$, which matches the…

We show that in the Lorentz mirror model, at any density of mirrors, the probability of a particle starting at the origin to reach distance n is at least 1/(2n+1).

This is the eighth and final paper in a series giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is…

We use 62185 quasars from the Sloan Digital Sky Survey (SDSS) DR5 sample to explore the quasar mass-luminosity plane view of virial mass estimation. Previous work shows deviations of ~0.4 dex between virial and reverberation masses. The…

The Szemer\'edi-Trotter theorem gives a bound on the maximum number of incidences between points and lines on the Euclidean plane. In particular it says that $n$ lines and $n$ points determine $O(n^{4/3})$ incidences. Let us suppose that an…

By comparing the results from numerical microlensing simulations to the observed long-term variability of quasars, strong upper limits on the cosmological density of compact objects in the 0.0001-1 solar mass range may in principle be…

In this paper we give an example of a closed, strongly one-sided dense set which is not of uniform density type. We also show that there is a set of uniform density type which is not of strong uniform density type.

This paper provides the currently best known upper bound on the density of a packing in three-dimensional Euclidean space of two types of spheres whose size ratio is the largest one that allows the insertion of a small sphere in each…

We establish a density variant of the Frankl-R\"{o}dl theorem on the sphere $\mathbb{S}^{n-1}$, which concerns avoiding pairs of vectors with a specific distance, or equivalently, a prescribed inner product. In particular, we establish…

We consider the problem of lower bounding a generalized Minkowski measure of subsets of a convex body with a log-concave probability measure, conditioned on the set size. A bound is given in terms of diameter and set size, which is sharp…

We show that a point set of cardinality $n$ in the plane cannot be the vertex set of more than $59^n O(n^{-6})$ straight-edge triangulations of its convex hull. This improves the previous upper bound of $276.75^n$.

For fixed integer $r\ge 2$, we call a pair $(m,f)$ of integers, $m\geq 1$, $0\leq f \leq \binom{m}{r}$, $absolutely$ $avoidable$ if there is $n_0$, such that for any pair of integers $(n,e)$ with $n>n_0$ and $0\leq e\leq \binom{n}{r}$ there…

We give a number of results about families of Ulam sets. Generalizing behavior of Ulam sets U(1,n), we prove using an novel model theoretic approach that there is a rigidity phenomenon for Ulam sets U(a,b) as b increases. Based on this, we…

Let $X$ be a Banach space. We study the circumstances under which there exists an uncountable set $\mathcal A\subset X$ of unit vectors such that $\|x-y\|>1$ for distinct $x,y\in \mathcal A$. We prove that such a set exists if $X$ is…

We prove the density hypothesis for wide families of arithmetic orbifolds arising from all division quaternion algebras over all number fields of bounded degree. Our power-saving bounds on the multiplicities of non-tempered representations…

Strong bounds are obtained for the number of automorphic forms for the group $\Gamma_0(q) \subseteq \operatorname{Sp}(4,\mathbb{Z})$ violating the Ramanujan conjecture at any given unramified place, which go beyond Sarnak's density…

The following generalisation of the Erd\H{o}s unit distance problem was recently suggested by Palsson, Senger and Sheffer. Given $k$ positive real numbers $\delta_1,\dots,\delta_k$, a $(k+1)$-tuple $(p_1,\dots,p_{k+1})$ in $\mathbb{R}^d$ is…

We study a generalization of Erd\H os's unit distances problem to chains of $k$ distances. Given $\mathcal P,$ a set of $n$ points, and a sequence of distances $(\delta_1,\ldots,\delta_k)$, we study the maximum possible number of tuples of…

We derived 90% confidence limits (CL) on the interstellar number density ($\rho_{IS}^{CL}$) of interstellar objects (ISO; comets and asteroids) as a function of the slope of their size-frequency distribution and limiting absolute magnitude.…