Ben Green and I have just uploaded to the arXiv our paper “New bounds for Szemeredi’s theorem, Ia: Progressions of length 4 in finite field geometries revisited“, submitted to Proc. Lond. Math. Soc.. This is both an erratum to, and a replacement for, our previous paper “New bounds for Szemeredi’s theorem. I. Progressions of length 4 in finite field geometries“. The main objective in both papers is to bound the quantity ${r_4(F^n)}$ for a vector space ${F^n}$ over a finite field ${F}$ of characteristic greater than ${4}$, where ${r_4(F^n)}$ is defined as the cardinality of the largest subset of ${F^n}$ that does not contain an arithmetic progression of length ${4}$. In our earlier paper, we gave two arguments that bounded ${r_4(F^n)}$ in the regime when the field ${F}$ was fixed and ${n}$ was large. The first “cheap” argument gave the bound

$\displaystyle r_4(F^n) \ll |F|^n \exp( - c \sqrt{\log n} )$

and the more complicated “expensive” argument gave the improvement

$\displaystyle r_4(F^n) \ll |F|^n n^{-c} \ \ \ \ \ (1)$

for some constant ${c>0}$ depending only on ${F}$.

Unfortunately, while the cheap argument is correct, we discovered a subtle but serious gap in our expensive argument in the original paper. Roughly speaking, the strategy in that argument is to employ the density increment method: one begins with a large subset ${A}$ of ${F^n}$ that has no arithmetic progressions of length ${4}$, and seeks to locate a subspace on which ${A}$ has a significantly increased density. Then, by using a “Koopman-von Neumann theorem”, ultimately based on an iteration of the inverse ${U^3}$ theorem of Ben and myself (and also independently by Samorodnitsky), one approximates ${A}$ by a “quadratically structured” function ${f}$, which is (locally) a combination of a bounded number of quadratic phase functions, which one can prepare to be in a certain “locally equidistributed” or “locally high rank” form. (It is this reduction to the high rank case that distinguishes the “expensive” argument from the “cheap” one.) Because ${A}$ has no progressions of length ${4}$, the count of progressions of length ${4}$ weighted by ${f}$ will also be small; by combining this with the theory of equidistribution of quadratic phase functions, one can then conclude that there will be a subspace on which ${f}$ has increased density.

The error in the paper was to conclude from this that the original function ${1_A}$ also had increased density on the same subspace; it turns out that the manner in which ${f}$ approximates ${1_A}$ is not strong enough to deduce this latter conclusion from the former. (One can strengthen the nature of approximation until one restores such a conclusion, but only at the price of deteriorating the quantitative bounds on ${r_4(F^n)}$ one gets at the end of the day to be worse than the cheap argument.)

After trying unsuccessfully to repair this error, we eventually found an alternate argument, based on earlier papers of ourselves and of Bergelson-Host-Kra, that avoided the density increment method entirely and ended up giving a simpler proof of a stronger result than (1), and also gives the explicit value of ${c = 2^{-22}}$ for the exponent ${c}$ in (1). In fact, it gives the following stronger result:

Theorem 1 Let ${A}$ be a subset of ${F^n}$ of density at least ${\alpha}$, and let ${\epsilon>0}$. Then there is a subspace ${W}$ of ${F^n}$ of codimension ${O( \epsilon^{-2^{20}})}$ such that the number of (possibly degenerate) progressions ${a, a+r, a+2r, a+3r}$ in ${A \cap W}$ is at least ${(\alpha^4-\epsilon)|W|^2}$.

The bound (1) is an easy consequence of this theorem after choosing ${\epsilon := \alpha^4/2}$ and removing the degenerate progressions from the conclusion of the theorem.

The main new idea is to work with a local Koopman-von Neumann theorem rather than a global one, trading a relatively weak global approximation to ${1_A}$ with a significantly stronger local approximation to ${1_A}$ on a subspace ${W}$. This is somewhat analogous to how sometimes in graph theory it is more efficient (from the point of view of quantative estimates) to work with a local version of the Szemerédi regularity lemma which gives just a single regular pair of cells, rather than attempting to regularise almost all of the cells. This local approach is well adapted to the inverse ${U^3}$ theorem we use (which also has this local aspect), and also makes the reduction to the high rank case much cleaner. At the end of the day, one ends up with a fairly large subspace ${W}$ on which ${A}$ is quite dense (of density ${\alpha-O(\epsilon)}$) and which can be well approximated by a “pure quadratic” object, namely a function of a small number of quadratic phases obeying a high rank condition. One can then exploit a special positivity property of the count of length four progressions weighted by pure quadratic objects, essentially due to Bergelson-Host-Kra, which then gives the required lower bound.