Related papers: More unit distances in arbitrary norms

As a variant of the celebrated Szemer\'edi--Trotter theorem, Guth and Katz proved that $m$ points and $n$ lines in $\mathbb{R}^3$ with at most $\sqrt{n}$ lines in a common plane must determine at most $O(m^{1/2}n^{3/4})$ incidences for…

We prove a common generalization to several mass partition results using hyperplane arrangements to split $\mathbb{R}^d$ into two sets. Our main result implies the ham-sandwich theorem, the necklace splitting theorem for two thieves, a…

We prove that there exist positive constants $C$ and $c$ such that for any integer $d \ge 2$ the set of ${\mathbf x}\in [0,1)^d$ satisfying $$ cN^{1/2}\le \left|\sum^N_{n=1}\exp\left (2 \pi i \left (x_1n+\ldots+x_d n^d\right)\right)…

Consider a set P of N random points on the unit sphere of dimension $d-1$, and the symmetrized set S = P union (-P). The halving polyhedron of S is defined as the convex hull of the set of centroids of N distinct points in S. We prove that…

A special case of a combinatorial theorem of De Bruijn and Erdos asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvatal suggested a possible generalization of this assertion in…

We study the problem of embedding bipartite graphs in Ahlfors-David regular sets of large dimension using results from extremal graph theory. Our main theorem states that any graph satisfying a power-improving bound on the extremal number…

We show that a coupling of non-colliding simple random walkers on the complete graph on $n$ vertices can include at most $n - \log n$ walkers. This improves the only previously known upper bound of $n-2$ due to Angel, Holroyd, Martin,…

The Ulam distance of two permutations on $[n]$ is $n$ minus the length of their longest common subsequence. In this paper, we show that for every $\varepsilon>0$, there exists some $\alpha>0$, and an infinite set $\Gamma\subseteq…

Let $\mathbb{F}_q$ be an arbitrary finite field, and $\mathcal{E}$ be a set of points in $\mathbb{F}_q^d$. Let $\Delta(\mathcal{E})$ be the set of distances determined by pairs of points in $\mathcal{E}$. By using the Kloosterman sums,…

By extending a construction due to Gross and McMullen [2], we show that for any odd integer n and for any even integer d>n+2 there are infinitely many Salem numbers $\alpha$ of degree d such that $\alpha^n-1$ is a unit. A similar result is…

We prove that any $n$ points in $\mathbb{R}^2$, not all on a line or circle, determine at least $\frac{1}{4}n^2-O(n)$ ordinary circles (circles containing exactly three of the $n$ points). The main term of this bound is best possible for…

We consider an analogue of the Kac random walk on the special orthogonal group $SO(N)$, in which at each step a random rotation is performed in a randomly chosen 2-plane of $\bR^N$. We obtain sharp asymptotics for the rate of convergence in…

We prove that if the Hausdorff dimension of $E\subset\mathbb{R}^d$, $d\geq 2$ is greater than $\frac{d}{2}+\frac{1}{3}$, the set of gaps of $2$-chains inside $E$, $$\Delta_2(E)=\{(|x-y|, |y-z|): x, y, z\in E \}\subset\mathbb{R}^2$$ has…

A subgraph $H$ of a graph $G$ is isometric if the distances between vertices in $H$ coincide with the distances between the corresponding vertices in $G$. We show that for any integer $n\ge 1$, there is a graph on $3^{n+O(\log^2 n)}$…

We show that the multiplicity of the second normalized adjacency matrix eigenvalue of any connected graph of maximum degree $\Delta$ is bounded by $O(n \Delta^{7/5}/\log^{1/5-o(1)}n)$ for any $\Delta$, and by…

We establish uniformization results for metric spaces that are homeomorphic to the euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of quasiconformality, we give a necessary and…

We prove that every $n$-vertex directed graph $G$ with the minimum outdegree $\delta^+(G) = d$ contains a subgraph $H$ satisfying \[ \min\left\{\delta^+(H), \delta^-(H) \right\} \ge \frac{d(d+1)}{2n} \,.\] We also show that if $d = o(n)$…

It was conjectured by Alon and proved by Friedman that a random $d$-regular graph has nearly the largest possible spectral gap, more precisely, the largest absolute value of the non-trivial eigenvalues of its adjacency matrix is at most…

We introduce a new type of distinct distances result: a lower bound on the number of distances between points on a line and points on a two-dimensional strip. This can be seen as a generalization of the well-studied problems of distances…

We bound the mean distance in a connected graph which is not a tree in function of its order $n$ and its girth $g$. On one hand, we show that mean distance is at most $\frac{n+1}{3}-\frac{g(g^2-4)}{12n(n-1)}$ if $g$ is even and at most…