Related papers: From Tarski's plank problem to simultaneous approx…

We give a deterministic, polynomial-time algorithm for approximately counting the number of {0,1}-solutions to any instance of the knapsack problem. On an instance of length n with total weight W and accuracy parameter eps, our algorithm…

The use of Hausdorff measures and dimension in the theory of Diophantine approximation dates back to the 1920s with the theorems of Jarnik and Besicovitch regarding well-approximable and badly-approximable points. In this paper we consider…

We use branes to generalize the Distance Conjecture. We conjecture that in any infinite-distance limit in the moduli space of a $d$-dimensional quantum gravity theory, among the set of particle towers and fundamental branes with at most…

We study the unit distance and distinct distances problems over the planar hypercomplex numbers: the dual numbers $\mathbb{D}$ and the double numbers $\mathbb{S}$. We show that the distinct distances problem in $\mathbb{S}^2$ behaves…

We study $L_2$-approximation problems $\text{APP}_d$ in the worst case setting in the weighted Korobov spaces $H_{d,\a,{\bm \ga}}$ with parameter sequences ${\bm \ga}=\{\ga_j\}$ and $\a=\{\az_j\}$ of positive real numbers $1\ge \ga_1\ge…

An important result in discrepancy due to Banaszczyk states that for any set of $n$ vectors in $\mathbb{R}^m$ of $\ell_2$ norm at most $1$ and any convex body $K$ in $\mathbb{R}^m$ of Gaussian measure at least half, there exists a $\pm 1$…

We introduce a new parameter, called stretch-width, that we show sits strictly between clique-width and twin-width. Unlike the reduced parameters [BKW '22], planar graphs and polynomial subdivisions do not have bounded stretch-width. This…

Suppose that $\langle f_n \rangle$ is a sequence of polynomials, $\langle f_n^{(k)}(0)\rangle$ converges for every non-negative integer $k$, and that the limit is not $0$ for some $k$. It is shown that if all the zeros of $f_1, f_2, \dots$…

We generalize the ham sandwich theorem for the case of well separated measures. Given convex bodies $K_1,...,K_d$ in $\mathbb{R_d}$ and numbers $\alpha_1,...,\alpha_d \in [0, 1]$, we give a sufficient condition for existence and uniqueness…

We prove that the subset sum problem has a polynomial time computable certificate of infeasibility for all $a$ weight vectors with density at most $1/(2n)$ and for almost all integer right hand sides. The certificate is branching on a…

In this note, we prove that smooth self-shrinkers in $\Real^{n+1}$, that are entire graphs, are hyperplanes. Previously Ecker and Huisken showed that smooth self-shrinkers, that are entire graphs and have at most polynomial growth, are…

Hadwiger's theorem is a Helly-type theorem involving common transversals to families of convex sets instead of common intersections. Subsequently, Pollack and Wenger identified a necessary and sufficient condition, called a consistent…

Suppose $A \in \mathbb{R}^{n \times n}$ is invertible and we are looking for the solution of $Ax = b$. Given an initial guess $x_1 \in \mathbb{R}$, we show that by reflecting through hyperplanes generated by the rows of $A$, we can generate…

The task of approximating a function of d variables from its evaluations at a given number of points is ubiquitous in numerical analysis and engineering applications. When d is large, this task is challenged by the so-called curse of…

We prove that for each $k\ge0$, the probability that a root vertex in a random planar graph has degree $k$ tends to a computable constant $d_k$, so that the expected number of vertices of degree $k$ is asymptotically $d_k n$, and moreover…

We prove the #P-hardness of the counting problems associated with various satisfiability, graph and combinatorial problems, when restricted to planar instances. These problems include \begin{romannum} \item[{}] {\sc 3Sat, 1-3Sat, 1-Ex3Sat,…

Let P_{n,d,D} denote the graph taken uniformly at random from the set of all labelled planar graphs on {1,2,...,n} with minimum degree at least d(n) and maximum degree at most D(n). We use counting arguments to investigate the probability…

We study the problem of enumerating Tarski fixed points on finite lattices. We derive query complexity lower bounds for finding three or more Tarski fixed points of isotone maps and the subclasses of increasing and decreasing isotone maps.…

We prove a new upper bound on the number of $r$-rich lines (lines with at least $r$ points) in a `truly' $d$-dimensional configuration of points $v_1,\ldots,v_n \in \mathbb{C}^d$. More formally, we show that, if the number of $r$-rich lines…

In the constrained planarity setting, we ask whether a graph admits a planar drawing that additionally satisfies a given set of constraints. These constraints are often derived from very natural problems; prominent examples are Level…