首页/论文/2011.08098

arXiv 2011.080982020-11math.CO

Distinct distances on non-ruled surfaces and between circles

Surya Mathialagan, Adam Sheffer

英文摘要：

We improve the current best bound for distinct distances on non-ruled algebraic surfaces in ${\mathbb R}^3$. In particular, we show that $n$ points on such a surface span $Ω\left(n^{32/39-\varepsilon}\right)$ distinct distances, for any $\varepsilon>0$. Our proof adapts the proof of Székely for the planar case, which is based on the crossing lemma. As part of our proof for distinct distances on surfaces, we also obtain new results for distinct distances between circles in ${\mathbb R}^3$. Consider point sets ${\mathcal P}_1$ and ${\mathcal P}_2$ of respective sizes $m$ and $n$, such that each set lies on a distinct circle in ${\mathbb R}^3$. We characterize the cases when the number of distinct distances between the two sets can be $O(m+n)$. This includes a new configuration with a small number of distances. In any other case, we prove that the number of distinct distances is $Ω\left(\min\left\{m^{2/3}n^{2/3},m^2,n^2\right\}\right)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Topologically Distinct Sets of Non-intersecting Circles in the Plane

Nested parentheses are forms in an algebra which define orders of evaluations. A class of well-formed sets of associated opening and closing parentheses is well studied in conjunction with Dyck paths and Catalan numbers. Nested parentheses also represent cuts through circles on a line. These become topologies of non-intersecting circles in the plane if the underlying algebra is commutative. This paper generalizes the concept and answers quantitatively - as recurrences and generating functions of matching rooted forests - the questions: how many different topologies of nested circles exist in t…

2016-09-19math.CO

Incidences between points and curves with almost two degrees of freedom

We study incidences between points and algebraic curves in three dimensions, taken from a family $C$ of curves that have almost two degrees of freedom, meaning that every pair of curves intersect in $O(1)$ points, for any pair of points $p$, $q$, there are only $O(1)$ curves of $C$ that pass through both points, and a pair $p$, $q$ of points admit a curve of $C$ that passes through both of them iff $F(p,q)=0$ for some polynomial $F$. We study two specific instances, one involving unit circles in $R^3$ that pass through some fixed point (so called anchored unit circles), and the other involving…

2020-06-23cs.CGmath.CO

Distinct Distances Between a Circle and a Generic Set

Let $S$ be a set of points in $\mathbb{R}^2$ contained in a circle and $P$ an unrestricted point set in $\mathbb{R}^2$. We prove the number of distinct distances between points in $S$ and points in $P$ is at least $\min(|S||P|^{1/4-\varepsilon},|S|^{2/3}|P|^{2/3},|S|^2,|P|^2)$. This builds on work of Pach and De Zeeuw, Bruner and Sharir, McLaughlin and Omar and Mathialagan on distances between pairs of sets.

2020-09-16math.COmath.MG

The Twelvefold way, the non-intersecting circles problem, and partitions of multisets

Let $n$ be a non-negative integer and $A=\{a_1,\ldots,a_k\}$ be a multi-set with $k$ not necessarily distinct members, where $a_1\leqslant\ldots\leqslant a_k$. We denote by $Δ(n,A)$ the number of ways to partition $n$ as the form $a_1x_1+\ldots+a_kx_k$, where $x_i$'s are distinct positive integers and $x_i< x_{i+1}$ whenever $a_i=a_{i+1}$. We give a recursive formula for $Δ(n,A)$ and some explicit formulas for some special cases. Using this notion we solve the non-intersecting circles problem which asks to evaluate the number of ways to draw $n$ non-intersecting circles in a plane regardless t…

2018-05-21math.CO

Few distinct distances implies no heavy lines or circles

We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set P of n points determines o(n) distinct distances, then no line contains Ω(n^{7/8}) points of P and no circle contains Ω(n^{5/6}) points of P. We rely on the bipartite and partial variant of the Elekes-Sharir framework that was presented by Sharir, Sheffer, and Solymosi in \cite{SSS13}. For the case of lines we combine this framework with a theorem from additive combinatorics, and for the case of circles we combine it with some basic algebraic geometry and a recen…

2013-08-26cs.CGmath.CO

Bisector energy and few distinct distances

We introduce the bisector energy of an $n$-point set $P$ in $\mathbb{R}^2$, defined as the number of quadruples $(a,b,c,d)$ from $P$ such that $a$ and $b$ determine the same perpendicular bisector as $c$ and $d$. If no line or circle contains $M(n)$ points of $P$, then we prove that the bisector energy is $O(M(n)^{\frac{2}{5}}n^{\frac{12}{5}+ε} + M(n)n^2).$. We also prove the lower bound $Ω(M(n)n^2)$, which matches our upper bound when $M(n)$ is large. We use our upper bound on the bisector energy to obtain two rather different results: (i) If $P$ determines $O(n/\sqrt{\log n})$ distinct dista…

2014-11-25math.CO

更多相关论文收起更多论文

Kronecker covers, V-construction, unit-distance graphs and isometric point-circle configurations

We call a polytope P of dimension 3 admissible if it has the following two properties: (1) for each vertex of P the set of its first-neighbours is coplanar; (2) all planes determined by the first-neighbours are distinct. It is shown that the Levi graph of a point-plane configuration obtained by V-construction from an admissible polytope P is the Kronecker cover of its 1-skeleton. We investigate the combinatorial nature of the V-construction and use it on unit-distance graphs to construct novel isometric point-circle configurations. In particular, we present an infinite series whose all members…

2012-07-29math.CO

Distinct distances between points and lines

We show that for $m$ points and $n$ lines in the real plane, the number of distinct distances between the points and the lines is $Ω(m^{1/5}n^{3/5})$, as long as $m^{1/2}\le n\le m^2$. We also prove that for any $m$ points in the plane, not all on a line, the number of distances between these points and the lines that they span is $Ω(m^{4/3})$. The problem of bounding the number of distinct point-line distances can be reduced to the problem of bounding the number of tangent pairs among a finite set of lines and a finite set of circles in the plane, and we believe that this latter question is o…

2015-12-28math.COmath.MG

Distinct distances on algebraic curves in the plane

Let $P$ be a set of $n$ points in the real plane contained in an algebraic curve $C$ of degree $d$. We prove that the number of distinct distances determined by $P$ is at least $c_d n^{4/3}$, unless $C$ contains a line or a circle. We also prove the lower bound $c_d' \min(m^{2/3}n^{2/3}, m^2, n^2)$ for the number of distinct distances between $m$ points on one irreducible plane algebraic curve and $n$ points on another, unless the two curves are parallel lines, orthogonal lines, or concentric circles. This generalizes a result on distances between lines of Sharir, Sheffer, and Solymosi in arXi…

2015-10-31math.COmath.MG

Exponential Sums and Distinct Points on Arcs

Suppose that some harmonic analysis arguments have been invoked to show that the indicator function of a set of residue classes modulo some integer has a large Fourier coefficient. To get information about the structure of the set of residue classes, we then need a certain type of complementary result. A solution to this problem was given by Gregory Freiman in 1961, when he proved a lemma which relates the value of an exponential sum with the distribution of summands in semi-circles of the unit circle in the complex plane. Since then, Freiman's Lemma has been extended by several authors. Rathe…

2008-12-26math.COmath.NT

On the Erdos distinct distance problem in the plane

In this paper, we prove that a set of $N$ points in ${\bf R}^2$ has at least $c{N \over \log N}$ distinct distances, thus obtaining the sharp exponent in a problem of Erdös. We follow the set-up of Elekes and Sharir which, in the spirit of the Erlangen program, allows us to study the problem in the group of rigid motions of the plane. This converts the problem to one of point-line incidences in space. We introduce two new ideas in our proof. In order to control points where many lines are incident, we create a cell decompostion using the polynomial ham sandwich theorem. This creates a dichotom…

2011-06-28math.CAmath.CO

确定圆的最小数量的简短证明

Short Proofs for the Minimum Number of Determined Circles

KU Leuven Campus Kulak-Kortrijk（荷语鲁汶大学库勒克-科特赖克校区）

本文针对Erdős问题506已完成解决方案的主要部分给出简短证明，通过反演论证、七点引理及结合Melchior不等式与Langer incidence bound的方法，确定了既非共线也非共圆的有限平面点集确定圆的最小数量。

2026-10-15math.COmath.HO

On polygon numbers of circle graphs and distance hereditary graphs

Circle graphs are intersection graphs of chords in a circle and $k$-polygon graphs are intersection graphs of chords in a convex $k$-sided polygon where each chord has its endpoints on distinct sides. The $k$-polygon graphs, for $k \ge 2$, form an infinite chain of graph classes, each of which contains the class of permutation graphs. The union of all of those graph classes is the class of circle graphs. The polygon number $\gp(G)$ of a circle graph $G$ is the minimum $k$ such that $G$ is a $k$-polygon graph. Given a circle graph $G$ and an integer $k$, determining whether $\gp(G) \le k$ is NP…

2017-10-04cs.DMmath.CO

加权限制中抛物面与球面的一个区别

A distinction between the paraboloid and the sphere in weighted restriction

该研究针对$\u211d^d$（$d\geq2$）中基于格点构造的权重，证明球面与抛物面的精确$L^2$加权限制不等式存在显著差异，运用泊松求和等方法证明，并推测一般稀疏权重下圆的加权限制效果更优。

2023-12-20math.CAmath.COmath.NT

On Bipartite Distinct Distances in the Plane

Given sets $\mathcal{P}, \mathcal{Q} \subseteq \mathbb{R}^2$ of sizes $m$ and $n$ respectively, we are interested in the number of distinct distances spanned by $\mathcal{P} \times \mathcal{Q}$. Let $D(m, n)$ denote the minimum number of distances determined by sets in $\mathbb{R}^2$ of sizes $m$ and $n$ respectively, where $m \leq n$. Elekes \cite{CircleGrids} showed that $D(m, n) = O(\sqrt{mn})$ when $m \leq n^{1/3}$. For $m \geq n^{1/3}$, we have the upper bound $D(m, n) = O(n/\sqrt{\log n})$ as in the classical distinct distances problem. In this work, we show that Elekes' construction is …

2019-12-04math.COmath.MG

三间隙定理的对称性

Symmetries of the Three Gap Theorem

本文研究了三间隙定理中的对称性，证明了间隙大小在单位圆上出现顺序的对称性结果。

2022-08-02math.COmath.DSmath.NT