Quantitative bounds for the U4-inverse theorem over low characteristic finite fields

- Department of Mathematics, Stanford University
- More about Jonathan Tidor

*Discrete Analysis*, October. https://doi.org/10.19086/da.38591.

### Editorial introduction

Quantitative bounds for the U4-inverse theorem over low characteristic finite fields, Discrete Analysis 2022:14, 17 pp.

Let G be a finite Abelian group and let f be a complex-valued function defined on G. For each a∈G let ∂af be the function given by the formula

∂af(x)=f(x)¯f(x−a).

Then for every integer k≥2 we can define a norm ‖.‖Uk on CG by the formula

‖f‖2kUk=Ex,a1,…,ak∂a1…∂akf(x),

where x,a1,…,ak are chosen uniformly and independently from G. It is not immediately obvious that this is a norm, but it can be shown to be with the help of multiple applications of the Cauchy-Schwarz inequality. (When k=1 the formula is still meaningful, but it simplifies to |Exf(x)|, which is not a norm but a seminorm, since it can equal zero for non-zero functions f.)

The Uk norms play an important role in additive combinatorics because they lead to a useful definition of quasirandomness for subsets of finite Abelian groups. For example, let k be a positive integer, let G be a finite Abelian group such that |G| has no non-trivial factors less than k, let A⊂G be a subset of density α, and for each x∈G, let f(x)=A(x)−α, where A(x)=1 if x∈A and 0 otherwise. Then for every ϵ>0 there exists c=c(ϵ,k)>0 such that if ‖f‖Uk≤ϵ, then

|Ex,dA(x)A(x+d)…A(x+(k−1)d)−αk|≤ϵ.

If A is a random subset of G of density α, then the expectation of Ex,dA(x)A(x+d)…A(x+(k−1)d) is very close to αk (the only reason we do not have equality is that when d=0 the events x+jd∈A are not independent), so the inequality above tells us that the number of arithmetic progressions in A of length k is roughly what it would be in a random set of the same density.

This fact focuses attention on what one can say about a function f if it takes values in [−1,1] but does *not* have a small Uk norm. When k=2, a straightforward argument shows that f must have a large Fourier coefficient, or equivalently that there is some character χ:G→C such that ⟨f,χ⟩ is large in absolute value. However, when k>2, the question becomes surprisingly difficult.

When G=Fnp for a prime p, and when p≥k, a result of Bergelson, Tao and Ziegler shows that f correlates with a polynomial phase function of degree at most k−1. That is, if f:Fnp→C is a function with |f(x)|≤1 for every x, and if ‖f‖Uk≥c>0, then there is a polynomial P in n variables of total degree at most k−1 such that |Exf(x)e−2πiP(x)/p|≥c′, where c′ is a positive constant that depends on c only.

This result, known as an *inverse theorem* for the Uk norm, was proved using infinitary methods, and therefore did not give an explicit dependence of c′ on c. Such a dependence was known when k=3, thanks to an earlier result of Green and Tao, and was obtained for general k by Gowers and Miličevič (though the bound they obtained is unlikely to be optimal).

Another direction in which the result was strengthened was to remove the condition that p≥k. It had been discovered by Green and Tao, and independently by Lovett, Meshulam, and Samorodnitsky, that the conclusion of the theorem was no longer true in general, and that it was necessary to introduce “non-classical polynomial phase functions” to obtain a correct statement. Tao and Ziegler followed up on the result described above with an inverse theorem in low characteristic, showing that a function with large Uk norm must correlate with a non-classical polynomial phase function.

However, it remains open to combine these two directions and prove a quantitative inverse theorem in the low-characteristic case. That is the problem addressed in this paper, which proves such a result when k=4 and sets out a clear programme for doing so for general k. The proof uses several elements of the proof of Gowers and Miličevič but departs from it at a point where the latter uses a polarization identity that requires dividing by (k−1)!.

An example of a non-classical polynomial can be constructed as follows. Consider first the map ϕ:F2→Z/4Z that sends 0 to 0 and 1 to 1. One can check that ϕ(x)−ϕ(x−a)=−a if x=0 and a if x=1, and that ϕ(x)−ϕ(x−a)−ϕ(x−b)+ϕ(x−a−b)=−2 if x=0,a=b=1, 2 if x=a=b=1, and 0 otherwise. Since −2=2 in Z/4Z, it follows that ϕ(x)−ϕ(x−a)−ϕ(x−b)+ϕ(x−a−b) does not depend on x. From this it follows that if we set g(x) to be iϕ(x), then for every a,b,c,x∈F2 we have that ∂a∂b∂cg(x)=1.

If we now define a function f:Fn2→C by f(x)=∏jg(xj), then ∂a∂b∂cf(x)=1 for every x,a,b,c∈Fn2, so ‖f‖U3=1. However, f is not of the form (−1)P for a quadratic function P:Fn2→F2. As it happens, Samorodnitsky has proved a quantitative U3 inverse theorem for functions defined on Fn2 that does not require non-classical polynomials, but in general they are needed. Thus, the result of this paper is the first quantitative inverse theorem for a case where non-classical polynomials necessarily appear.