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Suppose that each proper subset of a set $S$ of points in a vector space is contained in the union of planes of specified dimensions, but $S$ itself is not contained in any such union. How large can $|S|$ be? We prove a general upper bound…

We first describe a reduction from the problem of lower-bounding the number of distinct distances determined by a set $S$ of $s$ points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in…

We study the number of $k$-element sets $A \subset \{1,\ldots,N\}$ with $|A + A| \leq K|A|$ for some (fixed) $K > 0$. Improving results of the first author and of Alon, Balogh, Samotij and the second author, we determine this number up to a…

We consider the problem of minimizing a convex function over the intersection of finitely many simple sets which are easy to project onto. This is an important problem arising in various domains such as machine learning. The main difficulty…

Let $\Gamma$ be a cocompact discrete subgroup of $\mathrm{PSL}_{2}(\mathbb{C})$ and denote by $\mathcal{H}$ the three dimensional upper half-space. For a $p\in\mathcal{H}$, we count the number of points in the orbit $\Gamma p$, according to…

We obtain stringent bounds in the < r^2 >_S^{K pi}-c plane where these are the scalar radius and the curvature parameters of the scalar pi K form factor respectively using analyticity and dispersion relation constraints, the knowledge of…

In this note, we are interested in obtaining uniform upper bounds for the number of powerful numbers in short intervals $(x, x + y]$. We obtain unconditional upper bounds $O(\frac{y}{\log y})$ and $O(y^{11/12})$ for all powerful numbers and…

The metric sketching problem is defined as follows. Given a metric on $n$ points, and $\epsilon>0$, we wish to produce a small size data structure (sketch) that, given any pair of point indices, recovers the distance between the points up…

Motivated by a balanced ternary representation of the Collatz map we define the map $C_\mathbb{R}$ on the positive real numbers by setting $C_\mathbb{R}(x)=\frac{1}{2}x$ if $[x]$ is even and $C_\mathbb{R}(x)=\frac{3}{2}x$ if $[x]$ is odd,…

Let f(n) denote the smallest positive integer such that every set of $f(n)$ points in general position in the Euclidean plane contains a convex n-gon. In a seminal paper published in 1935, Erd\H{o}s and Szekeres proved that f(n) exists and…

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck^2/\log^* k pairs of parts which are not \epsilon-regular, where c,\epsilon>0 are absolute…

We collect some results in combinatorial geometry that follow from an inequality of Langer in algebraic geometry. Langer's inequality gives a lower bound on the number of incidences between a point set and its spanned lines, and was…

According to a classical result of Spencer, Szemer\'edi, and Trotter (1984), the maximum number of times the unit distance can occur among $n$ points in the plane is $O(n^{4/3})$. This is far from Erd\H{o}s's lower bound, $n^{1+O(1/\log\log…

Herein we prove an upper bound on the number of gravitationally lensed images in a generic multiplane point-mass ensemble with K planes and g_i masses in the ith plane. With E_K and O_K the sums of the even and odd degree terms respectively…

We develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for fixed $d \ge 1$, any subset of $[0, 1]^d$ of size $n$…

We use isoperimetric inequalities combined with a new technique to prove upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. In the process we prove tight new isoperimetric bounds for…

We verify the inequality $$ \frac{|K|}{|E|}+\frac{|K^*|}{|E^*|}\leq 2 $$ for any $o$-symmetric convex body $K\subset\mathbb{R}^2$ where $E$ is either the John ellipse of maximal area contained in $K$ or the minimal area L\"owner ellipse…

We generalize the Guth--Katz joints theorem from lines to varieties. A special case says that $N$ planes (2-flats) in 6 dimensions (over any field) have $O(N^{3/2})$ joints, where a joint is a point contained in a triple of these planes not…

An ordinary circle of a set $P$ of $n$ points in the plane is defined as a circle that contains exactly three points of $P$. We show that if $P$ is not contained in a line or a circle, then $P$ spans at least $\frac{1}{4}n^2 - O(n)$…

The aim of this paper is to prove that if a planar set $A$ has a difference set $\Delta(A)$ satisfying $\Delta(A)\subset \Z^++s$ for suitable $s$ than $A$ has at most 3 elements. This result is motivated by the conjecture that the disk has…