Pushing Behrend Around II

I will report on Christian Elsholtz, Zach Hunter, Laura Proske, Lisa Sauermann’s breakthrough paper Sets without arithmetic progressions in integers and over finite fields. The paper improves Behrend’s 1946 construction for 3-AP free sets of integers. An important aspect of the paper (and the entire field) is the interplay between questions over the integers and over finite fields.

I will also draw attention to Ben Green’s hidden yet widely known paper 100 open problems, and also to my own recent post Updates and plans  IV  where I mentioned quite a few mathematical breakthroughs.

Christian Elsholtz, Zach Hunter, Laura Proske, and Lisa Sauermann found remarkable improvement for Behrend’s construction.

Background

Erdos and Turan asked in 1936: What is the largest subset of {1,2,…,n} without a 3-term arithmetic progression? The size of is denoted by . The size of the largest subset of without 3-term AP is denoted by .

In 1946 Behrend found an example with In 2008, Michael Elkin pushed the the factor from the denominator to the enumerator, and found  a set with  

I wrote about Elkin’s result in one of my first blog posts. Since then we came back several times to upper bounds for and , and to related questions. Last year I wrote here about the exciting (for many reasons) Kelley-Meka upper bounds for .

The new result

 ,

where .

This is a major improvement over earlier results. Congratulations Christian, Zach, Laura, and Lisa.

Here is a link to the paper:

Improving Behrend’s construction: Sets without arithmetic progressions in integers and over finite fields by Christian Elsholtz, Zach Hunter, Laura Proske, and Lisa Sauermann

The question over finite fields and over the integers.

An important aspect in this area is the interplay between questions over the integers and analogous questions over finite fields. This connection is important for the new construction as well, and the constructions over finite fields are used for the constructions over the integers. This interplay is of course important for other problems as well: in 2008 Zeev Dvir settled the Kakeya problem over finite fields while the original question over the reals is still open. Drawing an even wider circle: Around 1940 Weil settled the Riemann Hypothesis (for curves) over finite fields, and the original RH is still open.

As for methods: Fourier methods work nicely in both cases (and are often easier in the finite field case). We do not know how to extend the polynomial method that sometimes do miracles in the finite field case to questions over the integers, and we do not have good methods to transfer results over finite fields to the integers (or, perhaps, a good formal understanding why we cannot expect it).

The Elsholtz et al.’s paper gives an improvement for , and thus also underlines that
the finite field model is very useful for the integer case. Another result from the paper is the following:

For every positive and large enough, , where is an absolute constant.

(Also here the value was a long standing threshold. There are two new main ingredients going into the proof (here I follow a personal communication with Christian):

a) The observation that a low-dimensional set in (here , see the drawing above) can circumvent the classical Behrend restriction of “at most half of the residue classes” and can be lifted to high dimension efficiently, to give , with .

b) This result in can be embedded into the integers.

What is the truth?

Here is the current state of knowledge

,

The exponent 1/9 is an improvement by Bloom and Sisask on the Kelly-Meka proof that gave the exponent 1/11. A common belief is that the true exponent is 1/2 (or ).  2.667… is the new bound that improves the earlier constant from Behrend’s construction . I don’t know if people regarded the constant from Behrend’s construction (and Elkin’s construction) as representing the truth. We had some public opinion poll regarding in this 2009 post . (See the results at the end of this post.) We welcome comments regarding people early and current beliefs on .

Ben Green’s list of 100 open problems

Ben Green wrote, by now a rather famous yet somewhat hidden, paper called “100 open problems“. Click on the title to reach the paper from Green’s personal site. The new Elsholtz et al.’s paper is already mentioned in Green’s paper.

Drawing attention to my post with short reports on many mathematical breakthroughs

Let me draw your attention to my recent post “Updates and plans IV” where I mentioned quite a few mathematical breakthroughs (divided into 17 “clusters”).  Clusters 7,11, and 16 describe results related to Roth’s and Szemeredi’s theorems and all the “cluster” are about remarkable mathematical developments that I would be happy blogging about in more details.

The results of our 2009 poll. The first option was removed already in 2010 by a theorem of Tom Sanders.  In 2020 Bloom and Sisask broke the logarithmic barrier. (The post describes some of the history.)  The Kelley-Meka’s breakthrough left us only with the two last options.

Update: Larger AP’s

Here is a comment by Zach Hunter.

Using these methods, you can always improve the constant in O’Bryant’s work from for explicit constants to (note: , not ; as a sanity check, the state-of-the-art is which correponds to in Behrend).

This improvement is not immediate, but if you are familiar with his construction, this can be done without too much effort by noting our building block and function satisfies a slightly more precise condition.

We did not include it in our paper, since I was already writing up a completely different argument which improves to . This will give an alternate proof of this improvement for and yields arbitrary improvements as (while the methods from the Behrend paper cannot be pushed to even get ).