# Incidences in Higher Dimensions: A Conjecture

I recently added to the incidence theory book two chapters about incidences in ${\mathbb R}^d$ (Chapters 8 and 9). At the moment incidence bounds in ${\mathbb R}^d$ are known for several different cases, and it is rather unclear how a general incidence bound in ${\mathbb R}^d$ should look like. In this post I would like to state what I believe this bound should look like. This conjecture is in part a result of conversations with Joshua Zahl over several years (I do not know whether Josh agrees with everything in this post). The post assumes some familiarity with incidence problems.

Joshua Zahl. A common name in this blog.

Let us start by stating some known bounds for incidences in ${\mathbb R}^d$. The first general incidence bound that relied on polynomial techniques was by Solymosi and Tao.

Theorem 1. Let ${\cal P}$ be a set of $m$ points and let ${\cal V}$ be a set of $n$ varieties, both in ${\mathbb R}^d$. The varieties of ${\cal V}$ are of degree at most $k$ and dimension at most $d/2$. The incidence graph of ${\cal P}\times {\cal V}$ contains no copy of $K_{s,t}$. Also, whenever two varieties $U_1,U_2 \in {\cal V}$ are incident to a point $p\in {\cal P}$, the tangent spaces of $U_1$ and $U_2$ at $p$ intersect at a single point. Then for any $\varepsilon>0$

$I({\cal P},{\cal V}) = O_{k,s,t,d,\varepsilon}\left(m^{\frac{s}{2s-1}+\varepsilon}n^{\frac{2s-2}{2s-1}}+m+n\right).$

While Theorem 1 provides a nice incidence bound, the varieties in it are somewhat restricted. A result of Fox et al. holds for varieties of any dimension and without the restriction on the tangent spaces, at the cost of having a weaker bound.

Theorem 2. Let $\cal P$ be a set of $m$ points and let $\cal V$ be a set of $n$ varieties of degree at most $k$, both in ${\mathbb R}^d$. Assume that the incidence graph of ${\cal P}\times {\cal V}$ contains no copy of $K_{s,t}$. Then for any $\varepsilon>0$

$I({\cal P},{\cal V}) = O_{k,s,t,d,\varepsilon}\left(m^{\frac{(d-1)s}{ds-1}+\varepsilon}n^{\frac{d(s-1)}{ds-1}}+m+n\right).$

Finally, a result of Sharir et al. provides a stronger bound for the special case of varieties of dimension one that do not cluster in a low degree surface (for brevity, the following is a simplified weaker variant).

Theorem 3. For every $\varepsilon>0$ there exists a constant $c_\varepsilon$ that satisfies the following. Let $\cal P$ be a set of $m$ points and let $\cal V$ be a set of $n$ irreducible varieties of dimension one and degree at most $k$, both in ${\mathbb R}^d$. Assume that the incidence graph of ${\cal P}\times {\cal V}$ contains no copy of $K_{s,t}$, and that every variety of dimension two and degree at most $c_\varepsilon$ contains at most $q$ elements of $\cal V$. Then

$I({\cal P},{\cal V}) = O_{k,s,t,d,\varepsilon}\left(m^{\frac{s}{ds-d+1} +\varepsilon}n^{\frac{ds-d}{ds-d+1}} + m^{\frac{s}{2s-1}+\varepsilon}n^{\frac{ds-d}{(d-1)(2s-1)}}q^{\frac{(s-1)(d-2)}{(d-1)(2s-1)}}+m+n\right).$

The incidence theory book describes in detail how to prove the three above bounds. With these result in mind, we are now ready to make a somewhat vague general conjecture.

Conjecture 4. Let $\cal P$ be a set of $m$ points and let $\cal V$ be a set of $n$ varieties of degree at most $k$ and dimension $d'$, both in ${\mathbb R}^d$. Assume that the incidence graph contains no copy of $K_{s,t}$ and that the varieties of $\cal V$ satisfy some “reasonable conditions”. Then for any $\varepsilon>0$

$I({\cal P},{\cal V}) = O_{k,s,t,d,\varepsilon}\left(m^{\frac{sd'}{ds-d+d'}+\varepsilon}n^{\frac{ds-d}{ds-d+d'}} + m + n\right).$

Note that we already have three cases of conjecture 4:

• Theorem 1 obtains the conjecture for varieties of dimension $d/2$.
• Theorem 2 obtains the conjecture for varieties of dimension $d-1$.
• Theorem 3 obtains the conjecture for varieties of dimension $1$.
For varieties of dimension smaller than $d/2$, the proof of Theorem 1 projects the configuration to a space of dimension that is twice the dimension of the varieties (forcing the dimension of the varieties to be exactly half the dimension of the space). This seems to be an inefficient step, which makes it likely that the theorem is not tight for varieties of dimension smaller than $d/2$. Indeed, Theorem 3 already obtains a stronger bound for the case of varieties of dimension one. For similar reasons, the proof of Theorem 2 seems not to be tight for varieties of dimension smaller than $d-1$.

A recent work reduces the distinct distances problem in ${\mathbb R}^d$ to an incidence problem between points and $(d-1)$-dimensional planes in ${\mathbb R}^{2d-1}$. Combining this reduction with the bound stated in Conjecture 4 would lead to an asymptotically tight bound for the distinct distances problem in dimension $d\ge3$. Other problems also reduce to incidence problems with the same dimensions. Thus, the case of $d' = (d-1)/2$ (for odd $d$) seems to be a main open case.

While conjecture 4 is known for the cases of varieties of dimension $1,d/2$, and $d-1$, so far all of the other cases are open. The bound in the conjecture was not obtained by interpolating the three known cases — there is a better approach that leads to it. Recall that in the polynomial partitioning technique we partition ${\mathbb R}^d$ into cells, bound the number of incidences in each cell, and then bound the number of incidences that are on the partition (not in any cell). The bound stated in Conjecture 4 is obtained by applying this technique while ignoring the incidences on the partition (since bounding the number of incidences in the cells is much easier).

Recently discovered configurations of points and varieties achieve the bound of the conjecture up to extra $\varepsilon$‘s in the exponents. Thus, in its most general form the bound of the conjecture is tight. However, these constructions give a tight bound only for certain families of varieties of dimension $d-1$, for certain ranges of $m$ and $n$, and when $s=2$. It seems plausible that the bound of the conjecture is not tight in some other interesting cases. In particular, in ${\mathbb R}^2$ a stronger bound is already known when $s\ge 3$, and it seems reasonable that the same technique would also work for varieties of dimension one in ${\mathbb R}^d$. At the moment, it is difficult to guess what should happen with varieties of dimension larger than one. Either way, Conjecture 4 seems to be a reasonable bound for the limits of the current polynomial partitioning technique. It seems that this specific technique could not yield better bounds without major changes.

Let us briefly discuss what the expression “reasonable conditions” in the statement of Conjecture 4 means. We already have two examples of conditions that are considered reasonable:

• A non-clustering condition: Not too many varieties of $\cal V$ are contained in a common higher-dimensional variety. For example, this restriction was used in Theorem 3, and more famously in the Guth–Katz proof of the planar distinct distances problem.
• A transversality condition: When two varieties meet at a point $p$, their tangent spaces at $p$ intersect only at one point. This condition was used in Theorem 1.
It is not clear what the “natural” restrictions are. For example, when studying incidences with two-dimensional planes in ${\mathbb R}^4$ most works rely on the transversality condition. In this case, the condition is equivalent to asking every two planes to intersect in at most one point. I recently noticed that the same incidence bound holds with a much weaker restriction of not too many planes in a hyperplane.

There is more to say about Conjecture 4, but this post is already getting longer than I intended.

# The k-set problem

The $k$-set problem is a classic open problem in Combinatorial Geometry, which is considered similar in spirit to incidence problems and to distance problems. While in recent years algebraic techniques led to significant progress in incidence and distance problems, so far none of the new methods were applied to the $k$-set problem. This post briefly introduces the problem, and describes a way of reducing it to an incidence problem. I hope that it will get someone to think of the $k$-set problem in an algebraic context and maybe come up with some new observations.

Let ${\cal P}$ be a set of $n$ points in ${\mathbb R}^2$, and let $1\le k\le n/2$ be an integer. A $k$-set of ${\cal P}$ is a subset ${\cal P}'\subset {\cal P}$ of $k$ points such that there exists a line that separates ${\cal P}'$ from ${\cal P}\setminus{\cal P}'$. That is, ${\cal P}'$ is on one open half-plane defined by the line, while ${\cal P}\setminus{\cal P}'$ is on the other half-plane. The following figure depicts two 5-sets of a set of 12 points.

The $k$-set problem asks for the maximum number of $k$-sets that a set of $n$-points in ${\mathbb R}^2$ can have. As a first silly example, note that the maximum number of 1-sets is exactly $n$. It is not difficult to show that we may assume that no three points are collinear (specifically, perturbing the point set cannot significantly decrease the number of $k$-sets).

The number of subsets of $k$ points of ${\cal P}$ is $\Theta(n^k)$, so this is a straightforward upper bound for the maximum number of $k$ sets. However, we can easily obtain a significantly stronger upper bound. Consider a $k$-set ${\cal P}'\subset{\cal P}$ that is separated by a line $\ell$. We translate $\ell$ until it is incident to a point $p\in {\cal P}'$ and then rotate $\ell$ around $p$ until it is incident to a second point of ${\cal P}$ (not necessarily in ${\cal P}'$). For example, see the figure below. Thus, to find all of the $k$-sets of ${\cal P}$ it suffices to consider every line that is incident to two points of ${\cal P}$. Since there are $O(n^2)$ such lines, this is also an upper bound for the number of $k$-sets.

# A First Draft of the Book “Incidence Theory”

You might have noticed that I did not post anything new for quite a while. The past months were unusually busy for me, due to personal reasons such as having my first child born(!), being on the job market, and several other things. I hope to return to my regular posting frequency around July.

The purpose of this post is to announce that I just uploaded the first draft of my book “Incidence Theory”. This book is about our current understanding of incidences (with a focus on the polynomial method), and their applications in other fields. I am trying to achieve two goals in this book: To have a clear and basic introduction of this subfield, while also creating a repository of results and techniques which may be used as a reference to experts. The current draft already contains several folklore results that I have not seen written before. It contains only the first seven chapters. I predict that the final version would contain about 15 chapters, and plan to gradually release the remaining ones.

Comments would be very appreciated, preferably by email. These can point out mistakes, typos, unclear formulations, suggestions for style changes, additional topics, simpler arguments, exercises, or anything else that might help improve the draft. The acknowledgements section is way too short. Please help me to extend it!

# Additive Energy of Real Point Sets

Over the years, more and more interactions between Discrete Geometry and Additive Combinatorics are being exposed. These include results such as the Green–Tao ordinary lines theorem and Solymosi’s sum-product bound. One reason for this connection is that both fields study the structure and symmetries of various objects (such as sets of points or subsets of additive groups). In this post I will discuss one of the simplest connections between the two fields — studying the additive energy of a set of points in a real space ${\mathbb R}^d$. The main goal of the post is to present two open problems that involve the additive energy of such sets. I heard one of these problems from Nets Katz and the other from Ciprian Demeter. In future posts we might discuss more involved interactions between the two fields.

Ciprian Demeter and Nets Katz.

# (More) Local Properties that imply Many Distinct Distances

Last year I wrote a post about a distinct distances problem that involves local properties of the point set. Specifically, let $\phi(n,k,l)$ denote the minimum number of distinct distances that can be determined by a set ${\cal P} \subset {\mathbb R}^2$ of $n$ points, such that any $k$ points of $\cal P$ determine at least $l$ distinct distances. That is, by assuming that the point set satisfies a local property, we wish to conclude the global property of many distinct distances. For more details and examples, see the original post.

A while ago I noticed another nice bound for this set of problems. Unfortunately the proof is very simple, which prevents it from being published in a “decent” journal. I’ve been trying to push it further, so far without success. Perhaps someone else would see how this idea can be extended.

As discussed in the previous post, Fox, Pach, and Suk proved that for every $k\ge 6$ and $\varepsilon>0$, we have

$\phi\left(n,k,\binom{k}{2}-k+6\right) = \Omega(n^{8/7-\varepsilon}).$

A simple argument shows that for every $k\ge 4$ we have

$\phi\left(n,k,\binom{k}{2}-\lfloor k/2 \rfloor +2\right) = \Omega(n^2).$

Our interest in this post lies in between these two bounds. That is, what happens between the values $\binom{k}{2}-k+6$ and $\binom{k}{2}- k/2 +2$. I am only aware of one previous result in thie range: Erdős and Gyárfás showed that $\phi\left(n,k,\binom{k}{2}-\lfloor k/2 \rfloor +1\right) = \Omega(n^{4/3})$. We will now prove a stronger result.

Theorem 1. For every $k\ge 8$ that is divisible by four, we have

$\phi\left(n,k,\binom{k}{2}- 3k/4 +3\right) = \Omega_k(n^{4/3}).$

# The Sum-Product Bound of Konyagin and Shkredov

In Solymosi’s famous 2009 paper, he proved that every finite set $A\subset {\mathbb R}$ satisfies

$|A+A||AA| = \Omega\left(|A|^{4/3}/\log^{1/3}|A|\right).$

In the past couple of years, Konyagin and Shkredov published two papers that extend Solymosi’s argument, obtaining a slightly stronger sum-product bound (one and two). These papers derive several additional results, and apply a variety of tools. I just uploaded to this blog my own exposition to the sum-product proof of Konyagin and Shkredov (a link can also be found in the pdf files page). This exposition ignores the additional results that are in the two papers, and tries to explain in detail every step that is part of the sum-product proof. In this aspect, the document would hopefully also fit beginners. As usual, I’m happy to receive any comments and corrections.

Ilya Shkredov and Sergei Konyagin.

# Additive Combinatorics Lecture Notes (part 2)

I finished my additive combinatorics class, and placed all of the lecture notes in the pdf files page. This quarter was rather short and I did not get to do several topics I had in mind. Perhaps I’ll add notes for some of these at some point. For now, let me just list the chapters that did not appear in the previous post.

• Chapter 4 is about arithmetic progressions in dense sets. It includes Behrend’s construction, Meshulam’s theorem, and Roth’s theorem. Due to the recent developments, Meshulam’s theorem already seems somewhat outdated…
• Chapter 5 consists of Sander’s proof of the quasi-polynomial Freiman-Ruzsa theorem in ${\mathbb F}_2^n$ (following Lovett’s presentation). This includes proving a special case of the probabilistic technique of Croot and Sisask.
• Chapter 6 presents the technique of relying on the third moment energy. It then uses this technique to study convex sets. I hope to also add a variant of the Balog-Szemerédi-Gowers theorem.