Good Bounds in Certain Systems of True Complexity One

Good bounds in certain systems of true complexity one, Discrete Analysis 2018:21, 40 pp.

In his analytic proof of Szemerédi’s theorem, Gowers introduced a sequence of norms, now known as the $U^k$ norms, given by the formula

$$\|f\|_{U^k}^{2^k}=\mathbb E_{x,a_1,\dots,a_k}\prod_{\epsilon\in\{0,1\}^k}C^{|\epsilon|}f\Bigl(x+\sum_i\epsilon_ia_i\Bigr),$$

where $f$ is a complex-valued function defined on a finite Abelian group, $C$ is the operation of complex conjugation and $|\epsilon|=\sum_i\epsilon_i$. For example,

$$\|f\|_{U^2}^4=\mathbb E_{x,a,b}f(x)\overline{f(x+a)f(x+b)}f(x+a+b).$$

A key lemma in his proof was that for any $k$ such functions $f_1,\dots,f_k$ that take values of modulus at most 1, one has the inequality

$$|\mathbb E_{x,d}f_1(x)f_2(x+d)\dots f_k(x+(k-1)d)|\leq\min_i\|f_i\|_{U^{k-1}}.$$

This lemma was proved using repeated applications of the Cauchy-Schwarz inequality. In such a situation we say that the average on the left-hand side is controlled by the $U^{k-1}$ norm.

Later, as part of their work on linear equations in the primes, Green and Tao considered more general expressions such as

$$\mathbb E_{x,y,z}f(x+y)f(y+z)f(x+z)f(x+y+z),$$

finding for each one a $k$ such that the expression is controlled by the $U^k$ norm. Their proof was a generalization of that of Gowers – again, it used repeated applications of the Cauchy-Schwarz inequality.

The $U^k$ norms increase with $k$, so if the average is controlled by the $U^k$ norm, then it is controlled by the $U^\ell$ norm for all $\ell>k$. It is therefore natural to ask what the minimal $k$ is that works for any given average of a product over linear forms. This question was examined in a series of papers of Gowers and Wolf, who conjectured that the average

$$\mathbb E_{x_1,\dots,x_r}f_1(L_1(x_1,\dots,x_r)\dots f_s(L_s(x_1,\dots,x_r))$$

is controlled by the $U^k$ norm if and only if the functions $L_1^k,\dots,L_s^k$ are linearly independent. To understand what this says, note that if the linear forms are $x, x+d, x+2d, x+3d$, then we have that $x^2-3(x+d)^2+3(x+2d)^2-(x+3d)^2=0$, so the $U^2$ norm does not control the average $\mathbb E_{x,d}f(x)f(x+d)f(x+2d)f(x+3d)$, but the cubes of the functions are linearly independent, so the $U^3$ norm does control it. Simple examples show that the independence condition is necessary, and in the case of arithmetic progressions, it agrees with the bound in the other direction that comes from the Cauchy-Schwarz argument. However, there are other systems of linear forms where the conjectured bound is strictly smaller than the bound proved by Green and Tao.

Gowers and Wolf proved their conjecture in many cases, and the remaining cases were proved by Green and Tao, and Hatami, Hatami and Lovett. All these proofs used difficult results known as inverse theorems for the $U^k$ norms, due to Bergelson, Tao and Ziegler, and Green, Tao and Ziegler. Since those theorems were known with only very poor bounds, the dependence of the averages on the $U^k$ norm given by these proofs was also very weak.

The situation is now improved, thanks largely to a very recent paper by the author of this one, but using the inverse theorems is still expensive. In this paper, however, it is shown, very surprisingly, that for products over six linear forms, it is possible to prove the result using the Cauchy-Schwarz inequality alone. This is in a sense the first interesting case of the theorem, since one needs six linear forms to obtain systems for which a straightforward Cauchy-Schwarz argument proves $U^3$ control, but in fact one has $U^2$ control.

As that last sentence suggests, the argument in this paper is anything but straightforward. Of course, the Cauchy-Schwarz inequality itself is not complicated, but the way the repeated applications are put together is highly ingenious. It is also shown in the paper that the ingenuity is in a certain sense necessary. As the author puts it, “the issue is that the Cauchy–Schwarz steps used must necessarily be tailor-made to the system Φ being considered. The task of describing a mapping from systems to Cauchy–Schwarz arguments could be likened to that of building a primitive computer using only the Cauchy–Schwarz inequality.” He demonstrates the necessity by proving that the bound genuinely depends not just on the number of linear forms but also on their coefficients. By contrast, the Cauchy-Schwarz argument in the paper of Green and Tao is uniform in the coefficients.