Distinct Distances Problem Research Articles

On Optimal Point Sets Determining Distinct Triangles

Erdős and Fishburn studied the maximum number of points in the plane that span $k$ distances (i.e. the set of pairwise distances between points has cardinality $k$) and classified these configurations, as an inverse problem of the Erdős distinct distances problem. We consider the analogous problem for triangles. Past work has obtained the optimal sets for one and two distinct triangles in the plane. In this paper, we resolve a conjecture that at most six points in the plane can span three distinct triangles, and obtain the hexagon as the unique configuration that achieves this. We also provide evidence that optimal sets cannot be on the square lattice in the general case.

A note on the distinct distances problem in the hyperbolic plane

A note on the distinct distances problem in the hyperbolic plane

Distinct Angles and Angle Chains in Three Dimensions

In 1946, Erd\H{o}s posed the distinct distance problem, which seeks to find the minimum number of distinct distances between pairs of points selected from any configuration of $n$ points in the plane. The problem has since been explored along with many variants, including ones that extend it into higher dimensions. Less studied but no less intriguing is Erd\H{o}s' distinct angle problem, which seeks to find point configurations in the plane that minimize the number of distinct angles. In their recent paper "Distinct Angles in General Position," Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolf use a logarithmic spiral to establish an upper bound of $O(n^2)$ on the minimum number of distinct angles in the plane in general position, which prohibits three points on any line or four on any circle. We consider the question of distinct angles in three dimensions and provide bounds on the minimum number of distinct angles in general position in this setting. We focus on pinned variants of the question, and we examine explicit constructions of point configurations in $\mathbb{R}^3$ which use self-similarity to minimize the number of distinct angles. Furthermore, we study a variant of the distinct angles question regarding distinct angle chains and provide bounds on the minimum number of distinct chains in $\mathbb{R}^2$ and $\mathbb{R}^3$.

Distinct Angle Problems and Variants

The Erdős distinct distance problem is a ubiquitous problem in discrete geometry. Less well known is Erdős’s distinct angle problem, the problem of finding the minimum number of distinct angles between n non-collinear points in the plane. The standard problem is already well understood. However, it admits many of the same variants as the distinct distance problem, many of which are unstudied. We provide upper and lower bounds on a broad class of distinct angle problems. We show that the number of distinct angles formed by n points in general position is $$O(n^{\log _{2}7})$$ providing the first non-trivial bound for this quantity. We introduce a new class of asymptotically optimal point configurations with no four cocircular points. Then, we analyze the sensitivity of asymptotically optimal point sets to perturbation, yielding a much broader class of asymptotically optimal configurations. In higher dimensions we show that a variant of Lenz’s construction admits fewer distinct angles than the optimal configurations in two dimensions. We also show that the minimum size of a maximal subset of n points in general position admitting only unique angles is $$\Omega (n^{1/5})$$ and $$O(n^{(\log _2{7)/3}})$$ . We also provide bounds on the partite variants of the standard distinct angle problem.

Distinct angles in general position

The Erdős distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is the Erdős distinct angle problem, the problem of finding the minimum number of distinct angles between n non-collinear points in the plane. Recent work has introduced bounds on a wide array of variants of this problem, inspired by analogous questions in the distance setting.In this short note, we improve the best known upper bound for the minimum number of distinct angles formed by n points in general position from O(nlog2⁡(7)) to O(n2). We consider a point-set to be in general position if no three points lie on a common line and no four lie on a common circle. Before this work, similar bounds relied on projections onto a generic plane from higher dimensional space. In this paper, we introduce a construction employing the geometric properties of a logarithmic spiral, sidestepping the need for a projection.We also apply this configuration to reduce the upper bound on the largest integer such that any set of n points in general position has a subset of that size with all distinct angles. This bound is decreased from O(nlog2⁡(7)/3) to O(n1/2).

On incidence bounds with Möbius hyperbolae in positive characteristic

We prove new incidence bounds between a plane point set, which is a Cartesian product, and a set of translates H of the hyperbola xy=λ≠0, over a field of asymptotically large positive characteristic p. They improve recent bounds by Shkredov, which are based on using explicit incidence estimates in the early terminated procedure of repeated applications of the Cauchy-Schwarz inequality, underlying many qualitative results related to growth and expansion in groups. The improvement – both quantitative, plus we are able to deal with a general H, rather than a Cartesian product – is mostly due to a non-trivial “intermediate” bound on the number of k-rich Möbius hyperbolae in positive characteristic. In addition, we make an observation that a certain energy-type quantity in the context of H can be bounded via the L2-moment of the Minkowski distance in H and can therefore fetch the corresponding estimates apropos of the Erdős distinct distance problem.

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 showed that $D(m, n) = O(\sqrt{mn})$ when $m \leqslant n^{1/3}$. For $m \geqslant n^{1/3}$, 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 tight by deriving the lower bound of $D(m, n) = \Omega(\sqrt{mn})$ when $m \leqslant n^{1/3}$. This is done by adapting Székely's crossing number argument. We also extend the Guth and Katz analysis for the classical distinct distances problem to show a lower bound of $D(m, n) = \Omega(\sqrt{mn}/\log n)$ when $m \geqslant n^{1/3}$.

Distinct distances with ℓp metrics

We study Erdős's distinct distances problem under ℓp metrics with integer p. We prove that, for every ε>0 and n points in R2, there exists a point that spans Ω(n6/7−ε) distinct distances with the other n−1 points. This improves upon the previous best bound of Ω(n4/5). We also characterize the sets that span an asymptotically minimal number of distinct distances under the ℓ1 and ℓ∞ metrics.

Note on the Number of Hinges Defined by a Point Set in ℝ2

It is shown that the number of distinct types of three-point hinges, defined by a real plane set of n points is ≫n2 log−3n, where a hinge is identified by fixing two pairwise distances in a point triple. This is achieved via strengthening (modulo a logn factor) of the Guth- Katz estimate for the number of pairwise intersections of lines in ℝ3, arising in the context of the plane Erdős distinct distance problem, to a second moment incidence estimate. This relies, in particular, on the generalisation of the Guth-Katz incidence bound by Solomon and Sharir.

Characterizing optimal point sets determining one distinct triangle

In this paper we determine the maximum number of points in $\mathbb{R}^d$ which form exactly $t$ distinct triangles, where we restrict ourselves to the case of $t = 1$. We denote this quantity by $F_d(t)$. It was known from the work of Epstein et al. that $F_2(1) = 4$. Here we show somewhat surprisingly that $F_3(1) = 4$ and $F_d(1) = d + 1$, whenever $d \geq 3$, and characterize the optimal point configurations. This is an extension of a variant of the distinct distance problem put forward by Erd\H{o}s and Fishburn.

A new bound for the Erdős distinct distances problem in the plane over prime fields

We obtain a new lower bound on the Erdős distinct distances problem in the plane over prime fields. More precisely, we show that for any set $A\subset \mathbb {F}_p^2$ with $|A|\le p^{7/6}$ and $p\equiv 3\mod 4$, the number of distinct distances deter

Distance sets over arbitrary finite fields

In this paper, we study the Erdős distinct distances problem for Cartesian product sets in the setting of arbitrary finite fields. More precisely, let Fq be an arbitrary finite field and A be a set in Fq. Suppose |A∩(aG)|≤|G|1/2 for any subfield G and a∈Fq∗, then |ΔFq(A2)|=|(A−A)2+(A−A)2|≫|A|1+121. Using the same method, we also obtain some results on sum–product type problems.

A reduction for the distinct distances problem in [formula omitted

We introduce a reduction from the distinct distances problem in Rd to an incidence problem with (d−1)-flats in R2d−1. Deriving the conjectured bound for this incidence problem (the bound predicted by the polynomial partitioning technique) would lead to a tight bound for the distinct distances problem in Rd. The reduction provides a large amount of information about the (d−1)-flats, and a framework for deriving more restrictions that these satisfy.Our reduction is based on introducing a Lie group that is a double cover of the special Euclidean group. This group can be seen as a variant of the Spin group, and a large part of our analysis involves studying its properties.

Schwartz-Zippel bounds for two-dimensional products

Schwartz-Zippel bounds for two-dimensional products, Discrete Analysis 2017:20, A famous open problem in combinatorial geometry is Erdős's unit-distances problem, which asks the following: given a subset $A\subset\mathbb R^2$ of size $n$, how many pairs $(a,b)\in A^2$ can there be with $d(a,b)=1$? (Here $d$ is the usual Euclidean distance.) The best-known upper bound, due to Spencer, Szemerédi and Trotter, is $O(n^{4/3})$, but it is conjectured that the true bound is $O(n^{1+\epsilon})$ for every $\epsilon>0$. (By putting all the points along a line, one can obtain a lower bound of $n-1$ with ease.) This problem is closely related to Erdős's distinct-distances problem, which was spectacularly solved by Guth and Katz, who showed that every set of size $n$ in $\mathbb R^2$ must give rise to at least $c_\epsilon n^{1-\epsilon}$ distinct distances. This would be a consequence of a positive answer to the unit-distances problem, since there are $n^2$ pairs of points and each distance would occur at most $C_\epsilon n^{1+\epsilon}$ times. An equally famous theorem in combinatorial geometry is the Szemerédi-Trotter theorem, which asserts that amongst any $n$ points and $m$ lines in $\mathbb R^2$, the number of incidences (that is, pairs $(P,L)$ where $P$ is one of the points, $L$ is one of the lines, and $P$ is on $L$) is at most $O(m+n+m^{2/3}n^{2/3})$. This paper concerns a simultaneous generalization of the Szemerédi-Trotter theorem and the upper bound above for the unit-distances problem. The connection is that both problems can be viewed as giving upper bounds for the size of the intersection of a variety with a Cartesian product of two finite subsets of the plane. In the case of the unit-distances problem, the two subsets are equal and the variety is the zero set of the function $f(a,b,c,d)=(a-c)^2+(b-d)^2-1$. The intersection of this zero set with the Cartesian product $A^2$ is the set of all pairs $((a,b),(c,d))$ such that $(a,b),(c,d)\in A$ and $d((a,b),(c,d))=1$, so the size of the intersection is the number of unit distances in $A$. As for the Szemerédi-Trotter theorem, if we let $P$ be the set of points, given by their Cartesian coordinates, and $L$ be the set of lines, associating the line $y=cx+d$ with the pair $(c,d)$, then the point $(a,b)$ belongs to the line $(c,d)$ if and only if $b=ca+d$, so the number of incidences is the size of the intersection of $P\times L$ with the zero set of the polynomial $ac+d-b$. This observation makes it tempting to conjecture that for every non-zero 4-variable polynomial $F$ and every pair $A,B$ of subsets of $\mathbb R^2$ of size $n$ there is an upper bound of $O(n^{4/3})$ on the intersection of $A\times B$ with the zero set of $F$. However, this is easily seen to be false. If $F$ has a formula of the form $F(x,y,s,t)=G(x,y)H(x,y,s,t)+K(s,t)L(x,y,s,t)$, then $A\times B$ is contained in the zero set of $F$ if $G$ vanishes on $A$ and $K$ vanishes on $B$. So some condition is needed on the polynomial for the conjecture to have a chance of being correct. The main result of this paper is essentially that the above source of examples is the only one: if a polynomial in four complex variables cannot be written in this way, then there is an upper bound of $C_\epsilon n^{4/3+\epsilon}$ for any $\epsilon>0$ for the intersection of its zero set with a Cartesian product of two subsets of $\mathbb C^2$ of size $n$. It is not clear whether this can be improved to an upper bound of $O(n^{4/3})$ -- hence the word "essentially" above. The title of the paper comes from the fact that this result can be viewed as a generalization of a special case of the Schwartz-Zippel lemma, which concerns products of subsets of $\mathbb C$ rather than subsets of $\mathbb C^2$. The proof depends on a two-dimensional generalization of a special case of yet another central result in combinatorial geometry, Alon's combinatorial Nullstellensatz. The paper also contains results about varieties of dimensions 1 and 2: for these the authors obtain a linear upper bound for the size of the intersection. The Szemerédi-Trotter theorem is known to give a tight bound, so in general the main result of the paper cannot be improved. However, this does not rule out improvements for specific polynomials. In the light of the unit-distances problem, it would naturally be of great interest to understand which polynomials might give rise to upper bounds that are better than $O(n^{4/3})$. One of the merits of this paper is that it focuses our attention on this question.

Distinct distances between points and lines

We show that for m points and n lines in R2, the number of distinct distances between the points and the lines is Ω(m1/5n3/5), as long as m1/2≤n≤m2. 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(m4/3). 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 of independent interest. In the same vein, we show that n circles in the plane determine at most O(n3/2) points where two or more circles are tangent, improving the previously best known bound of O(n3/2log⁡n). Finally, we study three-dimensional versions of the distinct point-line distances problem, namely, distinct point-line distances and distinct point-plane distances. The problems studied in this paper are all new, and the bounds that we derive for them, albeit most likely not tight, are non-trivial to prove. We hope that our work will motivate further studies of these and related problems.

Incidences Between Points and Lines on Two- and Three-Dimensional Varieties

Let P be a set of m points and L a set of n lines in \({\mathbb {R}}^4,\) such that the points of P lie on an algebraic three-dimensional variety of degree \(D\) that does not contain hyperplane or quadric components (a quadric is an algebraic variety of degree two), and no 2-flat contains more than s lines of L. We show that the number of incidences between P and L is $$\begin{aligned} I(P,L) = O\bigl (m^{1/2}n^{1/2}D + m^{2/3}n^{1/3}s^{1/3} + nD + m\bigr ) \end{aligned}$$for some absolute constant of proportionality. This significantly improves the bound of the authors (Sharir, Solomon, Incidences between points and lines in \({\mathbb {R}}^4.\) Discrete Comput Geom 57(3), 702–756, 2017), for arbitrary sets of points and lines in \({\mathbb {R}}^4,\) when \(D\) is not too large. Moreover, when \(D\) and s are constant, we get a linear bound. The same bound holds when the three-dimensional surface is embedded in any higher-dimensional space. The bound extends (with a slight deterioration, when \(D\) is large) to the complex field too. For a complex three-dimensional variety, of degree \(D,\) embedded in \({\mathbb {C}}^4\) (or in any higher-dimensional \({\mathbb {C}}^d\)), under the same assumptions as above, we have $$\begin{aligned} I(P,L) = O\bigl (m^{1/2}n^{1/2}D + m^{2/3}n^{1/3}s^{1/3} + D^6 + nD + m \bigr ). \end{aligned}$$For the proof of these bounds, we revisit certain parts of [36], combined with the following new incidence bound, for which we present a direct and fairly simple proof. Going back to the real case, let P be a set of m points and L a set of n lines in \({\mathbb {R}}^d,\) for \(d\ge 3,\) which lie in a common two-dimensional algebraic surface of degree \(D\) that does not contain any 2-flat, so that no 2-flat contains more than s lines of L (here we require that the lines of L also be contained in the surface). Then the number of incidences between P and L is $$\begin{aligned} I(P,L) = O\bigl (m^{1/2}n^{1/2}D^{1/2} + m^{2/3}D^{2/3}s^{1/3} + m + n\bigr ). \end{aligned}$$When \(d=3,\) this improves the bound of Guth and Katz (On the Erdős distinct distances problem in the plane. Ann Math 181(1), 155–190, 2015) for this special case, when \(D\ll n^{1/2}.\) Moreover, the bound does not involve the term O(nD). This term arises in most standard approaches, and its removal is a significant aspect of our result. Again, the bound is linear when \(D= O(1).\) This bound too extends (with a slight deterioration, when \(D\) is large) to the complex field. For a complex two-dimensional variety, of degree \(D,\) when the ambient space is \({\mathbb {C}}^3\) (or any higher-dimensional \({\mathbb {C}}^d\)), under the same assumptions as above, we have $$\begin{aligned} I(P,L) = O\bigl (m^{1/2}n^{1/2}D^{1/2} + m^{2/3}D^{2/3}s^{1/3} + D^3 + m + n\bigr ). \end{aligned}$$These new incidence bounds are among the very few bounds, known so far, that hold over the complex field. The bound for two-dimensional (resp., three-dimensional) varieties coincides with the bound in the real case when \(D= O(m^{1/3})\) (resp., \(D= O(m^{1/6})\)).

An improved point-line incidence bound over arbitrary fields

We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field $\mathbb{F}$, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that $m$ points and $n$ lines in $\mathbb{F}^2$, with $m^{7/8}<n<m^{8/7}$, determine at most $O(m^{11/15}n^{11/15})$ incidences (where, if $\mathbb{F}$ has positive characteristic $p$, we assume $m^{-2}n^{13}\ll p^{15}$). This improves on the previous best known bound, due to Jones. To obtain our bound, we first prove an optimal point-line incidence bound on Cartesian products, using a reduction to a point-plane incidence bound of Rudnev. We then cover most of the point set with Cartesian products, and we bound the incidences on each product separately, using the bound just mentioned. We give several applications, to sum-product-type problems, an expander problem of Bourgain, the distinct distance problem and Beck's theorem.

On the Erdős distinct distances problem in the plane

In this paper, we prove that a set of N points in R 2 has at least c N log N distinct distances, thus obtaining the sharp exponent in a problem of Erd} os. We follow the setup 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 decomposition using the polynomial ham sandwich theorem. This creates a dichotomy: either most of the points are in the interiors of the cells, in which case we immediately get sharp results or, alternatively, the points lie on the walls of the cells, in which case they are in the zero-set of a polynomial of suprisingly low degree, and we may apply the algebraic method. In order to control points incident to only two lines, we use the ecnode polynomial of the Rev. George Salmon to conclude that most of the lines lie on a ruled surface. Then we use the geometry of ruled surfaces to complete the proof.

On Range Searching with Semialgebraic Sets. II

Let $P$ be a set of $n$ points in $\mathbb{R}^d$. We present a linear-size data structure for answering range queries on $P$ with constant-complexity semialgebraic sets as ranges, in time close to $O(n^{1-1/d})$. It essentially matches the performance of similar structures for simplex range searching, and, for $d\ge 5$, significantly improves earlier solutions by the first two authors obtained in 1994. This almost settles a long-standing open problem in range searching. The data structure is based on a partitioning technique of Guth and Katz [On the Erdös distinct distances problem in the plane, arXiv:1011.4105, 2010], which shows that for a parameter $r$, $1 < r \le n$, there exists a $d$-variate polynomial $f$ of degree $O(r^{1/d})$ such that each connected component of $\mathbb{R}^d\setminus Z(f)$ contains at most $n/r$ points of $P$, where $Z(f)$ is the zero set of $f$. We present an efficient randomized algorithm for computing such a polynomial partition, which is of independent interest and is likely to have additional applications.

Simple Proofs of Classical Theorems in Discrete Geometry via the Guth–Katz Polynomial Partitioning Technique

Recently Guth and Katz ( arXiv:1011.4105, 2010) invented, as a step in their nearly complete solution of Erdős’s distinct distances problem, a new method for partitioning finite point sets in ℝ d , based on the Stone–Tukey polynomial ham-sandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemeredi–Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at self-contained, expository treatment. We also mention some generalizations and extensions, such as the Pach–Sharir bound on the number of incidences with algebraic curves of bounded degree.