Szemerédi’s theorem asserts that all subsets of the natural numbers of positive density contain arbitrarily long arithmetic progressions.  Roth’s theorem is the special case when one considers arithmetic progressions of length three.  Both theorems have many important proofs using tools from additive combinatorics, (higher order) Fourier analysis, (hyper) graph regularity theory, and ergodic theory.  However, the original proof by Endre Szemerédi, while extremely intricate, was purely combinatorial (and in particular “elementary”) and almost entirely self-contained, except for an invocation of the van der Waerden theorem.  It is also notable for introducing a prototype of what is now known as the Szemerédi regularity lemma.

Back in 2005, I rewrote Szemerédi’s original proof in order to understand it better, however my rewrite ended up being about the same length as the original argument and was probably only usable to myself.  In 2012, after Szemerédi was awarded the Abel prize, I revisited this argument with the intention to try to write up a more readable version of the proof, but ended up just presenting some ingredients of the argument in a blog post, rather than try to rewrite the whole thing.  In that post, I suspected that the cleanest way to write up the argument would be through the language of nonstandard analysis (perhaps in an iterated hyperextension that could handle various hierarchies of infinitesimals), but was unable to actually achieve any substantial simplifications by passing to the nonstandard world.

A few weeks ago, I participated in a week-long workshop at the American Institute of Mathematics on “Nonstandard methods in combinatorial number theory”, and spent some time in a working group with Shabnam Akhtari, Irfam Alam, Renling Jin, Steven Leth, Karl Mahlburg, Paul Potgieter, and Henry Towsner to try to obtain a manageable nonstandard version of Szemerédi’s original proof.  We didn’t end up being able to do so – in fact there are now signs that perhaps nonstandard analysis is not the optimal framework in which to place this argument – but we did at least clarify the existing standard argument, to the point that I was able to go back to my original rewrite of the proof and present it in a more civilised form, which I am now uploading here as an unpublished preprint.   There are now a number of simplifications to the proof.  Firstly, one no longer needs the full strength of the regularity lemma; only the simpler “weak” regularity lemma of Frieze and Kannan is required.  Secondly, the proof has been “factored” into a number of stand-alone propositions of independent interest, in particular involving just (families of) one-dimensional arithmetic progressions rather than the complicated-looking multidimensional arithmetic progressions that occur so frequently in the original argument of Szemerédi.  Finally, the delicate manipulations of densities and epsilons via double counting arguments in Szemerédi’s original paper have been abstracted into a certain key property of families of arithmetic progressions that I call the “double counting property”.

The factoring mentioned above is particularly simple in the case of proving Roth’s theorem, which is now presented separately in the above writeup.  Roth’s theorem seeks to locate a length three progression ${(P(1),P(2),P(3)) = (a, a+r, a+2r)}$ in which all three elements lie in a single set.  This will be deduced from an easier variant of the theorem in which one locates (a family of) length three progressions in which just the first two elements ${P(1), P(2)}$ of the progression lie in a good set (and some other properties of the family are also required).  This is in turn derived from an even easier variant in which now just the first element of the progression is required to be in the good set.

More specifically, Roth’s theorem is now deduced from

Theorem 1.5.  Let ${L}$ be a natural number, and let ${S}$ be a set of integers of upper density at least ${1-1/10L}$.  Then, whenever ${S}$ is partitioned into finitely many colour classes, there exists a colour class ${A}$ and a family ${(P_l(1),P_l(2),P_l(3))_{l=1}^L}$ of 3-term arithmetic progressions with the following properties:

1. For each ${l}$, ${P_l(1)}$ and ${P_l(2)}$ lie in ${A}$.
2. For each ${l}$, ${P_l(3)}$ lie in ${S}$.
3. The ${P_l(3)}$ for ${l=1,\dots,L}$ are in arithmetic progression.

The situation in this theorem is depicted by the following diagram, in which elements of $A$ are in blue and elements of $S$ are in grey:

Theorem 1.5 is deduced in turn from the following easier variant:

Theorem 1.6.  Let ${L}$ be a natural number, and let ${S}$ be a set of integers of upper density at least ${1-1/10L}$.  Then, whenever ${S}$ is partitioned into finitely many colour classes, there exists a colour class ${A}$ and a family ${(P_l(1),P_l(2),P_l(3))_{l=1}^L}$ of 3-term arithmetic progressions with the following properties:

1. For each ${l}$, ${P_l(1)}$ lie in ${A}$.
2. For each ${l}$, ${P_l(2)}$ and ${P_l(3)}$ lie in ${S}$.
3. The ${P_l(2)}$ for ${l=1,\dots,L}$ are in arithmetic progression.

The situation here is described by the figure below.

Theorem 1.6 is easy to prove.  To derive Theorem 1.5 from Theorem 1.6, or to derive Roth’s theorem from Theorem 1.5, one uses double counting arguments, van der Waerden’s theorem, and the weak regularity lemma, largely as described in this previous blog post; see the writeup for the full details.  (I would be interested in seeing a shorter proof of Theorem 1.5 though that did not go through these arguments, and did not use the more powerful theorems of  Roth or Szemerédi.)