Related papers: Unit Distances in Three Dimensions

Let $\gamma_1,\gamma_2$ be a pair of constant-degree irreducible algebraic curves in $\mathbb{R}^d$. Assume that $\gamma_i$ is neither contained in a hyperplane nor in a quadric surface in $\mathbb{R}^d$, for each $i=1,2$. We show that for…

Let $\mathfrak{L}$ be a collection of $L$ lines in $\R^3$ and $J$ the set of joints formed by $\mathfrak{L}$, i.e. the set of points each of which lies in at least 3 non-coplanar lines of $\mathfrak{L}$. It is known that $|J| \lesssim…

In 1984, H. W. Lenstra described an algorithm finding divisors of $N$ congruent to $r \mod S$. When $S^3 > N$, this algorithm runs in polynomial time and hence factors $N$ in time $N^{1/3+o(1)}$. Lenstra's algorithm relies on a sign change…

The Erd\H{o}s-Anning theorem states that every point set in the Euclidean plane with integer distances must be either collinear or finite. More strongly, for any (non-degenerate) triangle of diameter~$\delta$, at most $O(\delta^2)$ points…

We consider several problems that involve lines in three dimensions, and present improved algorithms for solving them. The problems include (i) ray shooting amid triangles in $R^3$, (ii) reporting intersections between query lines…

We study a variant of the Erd\H os unit distance problem, concerning angles between successive triples of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, and a sequence of angles…

In their 2009 note: \emph{Packing equal squares into a large square}, Chung and Graham proved that the uncovered area of a large square of side length $x$ is $O\left(x^{(3+\sqrt{2})/7}\log x\right)$ after maximum number of non-overlapping…

We show that if $A$ is a set of mutually orthogonal exponentials with respect to the unit disk then $|A \cap [-R, R]^2| \lesssim_\varepsilon R^{3/5+\varepsilon}$ holds. This improves the previous bound of $R^{2/3}$ by…

Metrics on the space of sets of trajectories are important for scientists in the field of computer vision, machine learning, robotics, and general artificial intelligence. However, existing notions of closeness between sets of trajectories…

In 1971 Trotter conjectured that every finite poset on at least $3$ points has a pair whose removal does not decrease the dimension by more than $1$. In 1992 Brightwell and Scheinerman introduced fractional dimension of posets, and they…

We survey the history and recent developments around two decades-old problems that continue to attract a great deal of interest: the slicing $\times 2$, $\times 3$ conjecture of H. Furstenberg in ergodic theory, and the distance set problem…

We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than $1$, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In…

We obtain new upper and lower bounds on the number of unit perimeter triangles spanned by points in the plane. We also establish improved bounds in the special case where the point set is a section of the integer grid.

We investigate the box dimensions of compact sets in $\mathbb{R}^2$ that contain a unit distance in every direction (such sets may have zero Hausdorff dimension). Among other results, we show that the lower box dimension must be at least…

We develop a new approach to address some classical questions concerning the size and structure of integer distance sets. Our main result is that any integer distance set in the Euclidean plane is either very sparse or has all but an…

In this paper, we use algorithmic tools, effective dimension and Kolmogorov complexity, to study the fractal dimension of distance sets. We show that, for any analytic set $E\subseteq\R^2$ of Hausdorff dimension strictly greater than one,…

Let $\{p_1, \ldots , p_n \} \subset {\Bbb{R}}^2$ be a separated point set, i.e., any two points have a distance at least $1$. Let $k \ge 1$ be an integer, and $1 \le t_1 < \ldots < t_k$ be real numbers. Let $\delta > 0$. Suppose for all $1…

In this paper, we provide upper bounds on several Rubinstein-type distances on the configuration space equipped with the Poisson measure. Our inequalities involve the two well-known gradients, in the sense of Malliavin calculus, which can…

Let $P$ be a finite point set in $\mathbb{R}^2$ with the set of distance $n$-chains defined as $$ \Delta_n(P)=\{(|p_1-p_2|,|p_2-p_3|,\ldots,|p_n-p_{n+1}|):p_i \in P\}.$$ We show that for $2\leq n=O_{|P|}(1)$ we have $$|\Delta_n(P)|\gtrsim…

In 2015, Guth proved that for any set of $k$-dimensional bounded complexity varieties in $\mathbb{R}^d$ and for any positive integer $D$, there exists a polynomial of degree at most $D$ whose zero set divides $\mathbb{R}^d$ into open…