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 . 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.
Last year I wrote a post about a distinct distances problem that involves local properties of the point set. Specifically, let denote the minimum number of distinct distances that can be determined by a set of points, such that any points of determine at least 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 and , we have
A simple argument shows that for every we have
Our interest in this post lies in between these two bounds. That is, what happens between the values and . I am only aware of one previous result in thie range: Erdős and Gyárfás showed that . We will now prove a stronger result.
Theorem 1. For every that is divisible by four, we have
In Solymosi’s famous 2009 paper, he proved that every finite set satisfies
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.
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 (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.
Andrew Suk just placed on arXiv a paper with an almost tight bound for the happy ending problem. These are exciting news for discrete geometers! I’ll use the opportunity to describe the problem and its non-technical history.
Andrew Suk and the Anonymous statue.
In the early 1930’s in Budapest, several students used to regularly meet on Sundays in a specific city park bench. This was the bench next to the statue of Anonymous, the first medieval Hungarian chronicler. Among these students were Paul Erdős, George Szekeres, and Esther Klein.
On one such Sunday meeting, Klein pointed out that for any set of five points with no three on a line, four of the points are the vertices of a convex quadrilateral. This easy observation can be proved by considering the convex hull of the five points. If at least four of the points are on the boundary of the convex hull, then we are done. If the convex hull consists only of three of the points, then we consider the line that passes through the two interior points. We take the two interior points and the two points that are on the same side of .
The Budapest students started to think about whether there exists an such that every set of points with no three on a line contains the vertices of a convex pentagon. More generally, does a similar condition hold for every convex -gon? The first to prove the claim was Szekeres. A couple of years later, Esther Klein, who suggested the problem married George Szekeres who solved it. Since then, this problem is called the happy ending problem. They lived together up to their 90’s.
For every integer , let denote the smallest integer such that any set of points with no three on a line contains a subset of points that are the vertices of a convex -gon. Erdős and Szekeres published a paper on the problem in 1935. They proved that
Erdős and Szekeres conjectured that , and this is sometimes called “the Erdős–Szekeres conjecture”. Erdős offered a 500$ ` for proving or disproving this conjecture, and a few years ago Ron Graham (who is handling Erdős’ problem awards) increased the amount to 1000$. So far the conjecture was verified up to by a computer. Over the years several improved bounds were published, but the improvements were always sub-exponential. That is, the value of remained between and (ignoring sub-exponential factors). In his new paper, Suk derives the impressive bound , settling the problem up to sub-exponential factors. Surprisingly, after 81 years of small progress, Suk’s impressive improvement requires a short proof that does not seem to rely on any kind of heavy machinery.
I recently started teaching an “Additive combinatorics” class, and am writing lecture notes for it. So far I put the first three chapters online. I’d appreciate any comments about these, from pointing out serious mistakes, to pointing out minor typos, or even a recommendation for the final topic (which I have not chosen yet). The chapters that are already up are:
- In Chapter 1 we start to study the principle that sets with small doubling must have structure. We prove some basic results such as Ruzsa’s triangle inequality, Plünnecke’s inequality, and variants of Freiman’s theorem.
- Chapter 2 studies the sum-product problem over the reals. In addition to showing the proofs of Elekes and Solymosi, we see how the same techniques can be applied to several other problems.
- Chapter 3 discusses the The Balog-Szemerédi-Gowers theorem. Specifically, we present the variant of Schoen and the variant by Sudakov, Szemerédi, and Vu.
As the quarter progresses, I will keep uploading more chapters.
This post is about a distinct distances problem which I think is quite interesting. As we shall see, solving this problem would also lead to significant progress in characterizing point sets with a small number of distinct distances. Recall that a perpendicular bisector of two points is the line that consists of the points satisfying (where is the distance between and ). For brevity, we will simply refer to these as bisectors.
We say that a set of points in has no repeated bisectors if for every four distinct points the bisector of and is not identical to the bisector of and (that is, the configuration in the above figure is forbidden).
Conjecture. Let be a set of points in with no repeated bisectors. Then for any the points of determine distinct distances, .
For some initial intuition, consider the point sets that determine a small number of distinct distances. I am only aware of two types of configurations that yield distinct distances: the vertices of a regular -gon and lattices.
We can also play a bit with these examples. For example, by taking two regular -gons whose vertices are on two concentric circles, or by taking evenly spaced points on a line (that is, an lattice). All of these examples have a significantly large amount of repeated bisectors.
So far I only managed to find a set with no repeated bisectors and distinct distances. From the other direction, I can only show that every such set determines distinct distances. This is unfortunate, since a stronger bound for this problem would lead to a breakthrough in characterizing planar point sets that determine a small number of distinct distances. Below I describe how this can be done, and also how to obtain simple lower and upper bounds for this problem.