A Multidimensional Szemerédi Theorem in the Primes via Combinatorics

Abstract

Let A be a subset of positive relative upper density of $\mathbb {P}^d$, the d-tuples of primes. We present an essentially self-contained, combinatorial argument to show that A contains infinitely many affine copies of any finite set $F\subseteq \mathbb {Z}^d$. This provides a natural multidimensional extension of the theorem of Green and Tao on the existence of long arithmetic progressions in the primes.

Appendix A: Basic Properties of Weighted Box Norms

In this appendix, we describe some basic facts about the weighted version of Gowers’s box norms defined in (1.29) for functions $F:V_e\rightarrow \mathbb {R}$. We will assume $e=\{1,\ldots ,d\}=:[d]$, and $V=V_{[d]}=\mathbb {Z}_N^d$ without loss of generality. To show that these are indeed norms (for $d\ge 2$) let us define a multilinear form referred to as the weighted Gowers’s inner product. Let $F_\omega :V_e\rightarrow \mathbb {R}$ for $\omega \in \{0,1\}^e$, be a given family of functions and define

$$\begin{aligned}&\left\langle F_{\omega };\omega \in \{0,1\}^d \right\rangle _{\Box _{\nu }}\\&\quad := \mathbb {E}_{x_{[d]},y_{[d]}\in V}\prod _{\omega \in \{0,1\}^d} F_{\omega }(\omega (x_{[d]},y_{[d]}))\prod _{|I|<d}\prod _{\omega _I\in \{0,1\}^I} \nu _I(\omega _I(x_I,y_I)). \end{aligned}$$

Therefore, $\left\langle F_{\omega };\omega \in \{0,1\}^d \right\rangle _{\Box _{\nu }}=\left\| F\right\| _{\Box _{\nu }}^{2^d}$, if $F_\omega =F$ for all $\omega \in \{0,1\}^e$.

Lemma A.1

(The Gowers–Cauchy–Schwartz inequality). $|\left\langle F_{\omega }; \omega \in \{0,1\}^d \right\rangle | \le { \prod _{\omega _{[d]}} } \left\| F_{\omega }\right\| _{\Box ^d_{\nu }}.$

Proof

We will use the Cauchy–Schwartz inequality several times and the linear form condition. We have

$$\begin{aligned}&\left\langle F_{\omega } ; \omega \in \{0,1\}^d \right\rangle _{\Box _{\nu }^d}\\&\quad =\mathbb {E}_{x_{[2,d]},y_{[2,d]}}\left( \left( \prod _{|I|<d,1 \notin I} \prod _{\omega _I} \nu _I(\omega _I(x_I,y_I)) \right) ^{1/2} \right. \\&\qquad \quad \times \left( \mathbb {E}_{x_1} \nu (x_1) \prod _{\omega _{[2,d]}} F_{\omega _{(0,[2,d])}}(x_1,\omega _{[2,d]}(x_{[2,d]},y_{[2,d]}))\right) \\&\qquad \quad \times \prod _{|I|<d-1,1 \notin I}\nu _{\{1\} \cup I}(x_1,\omega _I(x_I,y_I)) \left( \prod _{|I|<d,1 \notin I} \prod _{\omega _I} \nu _I({\omega _I}(x_I,y_I)) \right) ^{1/2} \\&\qquad \quad \times \left( \mathbb {E}_{y_1} \nu (y_1) \prod _{\omega _{[2,d]}} F_{\omega _{(1,[2,d])}}(y_1,{\omega _{[2,d]}}(x_{[2,d]},y_{[2,d]}))\right) \nonumber \\&\qquad \quad \left. \times \prod _{|I|<d-1,1 \notin I}\nu _{\{1\} \cup I}(y_1,{\omega _I}(x_I,y_I))\right) . \end{aligned}$$

Applying the Cauchy–Schwartz inequality in the $x_1$ variable, one has

$$\begin{aligned} \left| \left\langle F_{\omega } ; \omega \in \{0,1\}^d \right\rangle _{\Box _{\nu }^d} \right| ^2 \le A \cdot B \end{aligned}$$

here,

$$\begin{aligned} A&=\mathbb {E}_{x_{[2,d]},y_{[2,d]}} \left( \prod _{|I|<d,1 \notin I} \prod _{\omega _I} \nu _I({\omega _I}(x_I,y_I))\right. \\&\quad \quad \times \mathbb {E}_{x_1,y_1}\nu (x_1)\nu (y_1) \prod _{\omega _{[2,d]}} F_{\omega _{(0,[2,d])}}(x_1,{\omega _{[2,d]}}(x_{[2,d]},y_{[2,d]}))\\&\quad \quad \times \, F_{\omega _{(0,[2,d])}}(y_1,{\omega _{[2,d]}}(x_{[2,d]},y_{[2,d]})) \\&\quad \quad \left. \times \prod _{|I|<d-1,1 \notin I}\nu _{\{1\} \cup I}(x_1,{\omega _I}(x_I,y_I)) \right) = \left\langle F^{(0)}_{\omega }({\omega }(x_{[d]},y_{[d]})) \right\rangle _{\Box ^d_{\nu }}, \end{aligned}$$

where

$$\begin{aligned} F^{(0)}_{(0,\omega _{[2,d]})}(x_1,{\omega _{[2,d]}}(x_{[2,d]},y_{[2,d]}))&= F^{(0)}_{(1,\omega _{[2,d]})}(y_1,{\omega _{[2,d]}}(x_{[2,d]},y_{[2,d]}))\\&:=F(x_1,{\omega _{[2,d]}}(x_{[2,d]},y_{[2,d]})) \end{aligned}$$

for any $\omega _{[2,d]}$. Similarly

$$\begin{aligned} B =\left\langle F^{(1)}_{\omega }({\omega }(x_{[d]},y_{[d]})) \right\rangle _{\Box ^d_{\nu }}, \end{aligned}$$

where

$$\begin{aligned} F^{(1)}_{(0,\omega _{[2,d]})}(x_1,{\omega _{[2,d]}}(x_{[2,d]},y_{[2,d]}))&= F^{(1)}_{(1,\omega _{[2,d]})}(y_1,{\omega _{[2,d]}}(x_{[2,d]},y_{[2,d]}))\\&:=F(y_1,{\omega _{[2,d]}}(x_{[2,d]},y_{[2,d]})) \end{aligned}$$

for any $\omega _{[2,d]}.$ In the same way, applying the Cauchy–Schwartz inequality in $x_2$ variable, we end up with

$$\begin{aligned} |\left\langle F_{\omega }; \omega \in \{0,1\}^d \right\rangle _{\Box ^d_{\nu }}| \le \prod _{\omega _{[0,1]}}\left\langle F^{\omega _{[0,1]}}_{\omega }; \omega \in \{0,1\}^d \right\rangle _{\Box ^d_{\nu }} \end{aligned}$$

and continuing this way with $x_3,\ldots ,x_d$ variables, we end up with

$$\begin{aligned} |\left\langle F_{\omega }; \omega \in \{0,1\}^d \right\rangle _{\Box ^d_{\nu }}| \le \prod _{\omega \in \{0,1\}^d}\left\langle F_\omega ,\ldots ,F_\omega \right\rangle _{\Box ^d_{\nu }} = \prod _{\omega \in \{0,1\}^d} \left\| F_{\omega } \right\| ^{2^d}_{\Box ^d_{\nu }}. \end{aligned}$$

$\square$

Corollary A.1

$\left\| \cdot \right\| _{{\Box ^d_{\nu }}}$ is a semi-norm for $d\ge 1$.

Proof

By the Gowers–Cauchy–Schwartz inequality, we have that $\Vert F\Vert _{\Box _\nu }\ge 0$; moreover

$$\begin{aligned} \left\| F+G \right\| _{\Box _{\nu }^d}^{2^d}&= \left\langle F+G,\ldots ,F+G \right\rangle _{\Box _{\nu }^d} \\&=\sum _{\omega \in \{0,1\}^d}\left\langle h^{\omega _1},\ldots ,h^{\omega _d} \right\rangle _{\Box _{\nu }^d},\quad h^{\omega }={\left\{ \begin{array}{ll} F, &{}\quad \omega =0 \\ G, &{}\quad \omega =1 \end{array}\right. }\\&\le \sum _{\omega \in \{0,1\}^d} \left\| h^{\omega _1} \right\| _{\Box _{\nu }^d}\ldots \left\| h^{\omega _d} \right\| _{\Box _{\nu }^d} =(\left\| F \right\| _{\Box _{\nu }^d}+\left\| G \right\| _{\Box _{\nu }^d})^{2^d}. \end{aligned}$$

In addition, it follows directly from the definition that $\left\| \lambda F\right\| _{\Box _{\nu }^d}^{2^d}=\lambda ^{2^d}\left\| f \right\| _{\Box _{\nu }^d}^{2^d}$; hence, $\left\| \lambda F \right\| _{\Box _{\nu }^d}=|\lambda |\left\| F \right\| _{\Box _{\nu }^d}.$ $\square$

Proof of Proposition 1.1

Let $\mathcal {H}'=\{f\in \mathcal {H};\ |f|<d$}, and write the left side of (1.26) as

$$\begin{aligned} \mathbb {E}=\mathbb {E}_{x\in V_J}\prod _{e\in \mathcal {H}_d} F_e(x_e)\prod _{f\in \mathcal {H}'}\nu _f(x_f). \end{aligned}$$

Fix $e_0=[d]$ and write $e_j:=[d+1]\backslash \{j\}$ for the rest of the faces. The idea is to apply the Cauchy–Schwartz inequality successively in the $x_1,x_2,\ldots ,x_d$ variables to eliminate the functions $F_{e_1}\le \nu _{e_1},\ldots ,F_{e_d}\le \nu _{e_d}$, using the linear form condition at each step. Using $F_{e_1}\le \nu _{e_1}$, we have

$$\begin{aligned} |E|\le \mathbb {E}_{x_2,\ldots ,x_{d+1}} \nu _{e_1}(x_1)\prod _{1\notin f\in \mathcal {H}'} \nu _f(x_f)\left| \mathbb {E}_{x_1}\prod _{j\ne 2}F_{e_j}(x_j)\prod _{1\in f\in \mathcal {H}'}\nu _f(x_f)\right| . \end{aligned}$$

By the linear form condition $\mathbb {E}_{x_2,\ldots ,x_{d+1}} \nu _{e_1}(x_1)\prod _{1\notin f\in \mathcal {H}'} \nu _f(x_f)=1+o_{N \rightarrow \infty }(1)$, thus by the Cauchy–Schwartz inequality,

$$\begin{aligned} E^2&\lesssim \mathbb {E}_{x_2,\ldots ,x_{d+1}} \nu _{e_1}(x_1)\prod _{1\notin f\in \mathcal {H}'} \nu _f(x_f)\ \mathbb {E}_{x_1,y_1} \prod _{j\ne 2}F_{e_j}(x_1,x_{e_j\backslash \{1\}})F_{e_j}(y_1,x_{e_j\backslash \{1\}})\nonumber \\&\quad \times \, \prod _{1\in f\in \mathcal {H}'}\nu _f(y_1,x_{f\backslash \{1\}})\,\nu _f(x_1,x_{f\backslash \{1\}}). \end{aligned}$$

(A.1)

Note that, what happened is that we have replaced the function $F_{e_1}$ by the measure $\nu _{e_1}$, doubled the variable $x_1$ to the pair of variables $(x_1,y_1)$ and also doubled each factor of the form $G_e(x_e)$ (which is either $F_e(x_e)$ or $\nu _e(x_e)$, for $e\in \mathcal {H}$) depending on the $x_1$ variable. To keep track of these changes as we continue with the rest of that variables, let us introduce some notations. Let $g\subseteq [d]$ and for a function $G_e(x_e)$ define

$$\begin{aligned} G_e^*(x_{e\cap g},y_{e\cap g},x_{e\backslash g}) := \prod _{\omega _e\in \{0,1\}^{e\cap g}} G_e(\omega _e(x_{e\cap g},y_{e\cap g}),x_{e\backslash g}). \end{aligned}$$

(A.2)

We claim that after applying the Cauchy–Schwartz inequality in the $x_1,\ldots ,x_i$ variables, we have with $g=[i]$

$$\begin{aligned} E^{2^i}&\lesssim \mathbb {E}_{x_{[i]},y_{[i]},x_{J\backslash [i]}} \prod _{j\le i} \nu _{e_j}^*(x_{[i]\cap e_j},y_{[i]\cap e_j},x_{e_j\backslash [d]}) \prod _{j>i} F_{e_j}^* (x_{[i]\cap e_j}, y_{[i]\cap e_j}, x_{e_j\backslash [d]})\nonumber \\&\quad \times \prod _{f\in \mathcal {H}'} \nu _f^* (x_{f\cap [i]},y_{f\cap [i]},x_{f\backslash [i]}). \end{aligned}$$

(A.3)

For $i=1$, this can be seen from (A.1). Note that the linear forms appearing in any of these factors are pairwise linearly independent as our system is well-defined. Assuming it holds for i separating the factors independent of the $x_{i+1}$ variable, replacing the function $F_{e_{i+1}}$ with $\nu _{e_{i+1}}$, and applying the Cauchy–Schwartz inequality we double the variable $x_{i+1}$ to the pair $(x_{i+1},y_{i+1})$ and each factor $G^*_e(x_{e\cap [i]},y_{e\cap [i]},x_{e\backslash [i]})$ depending on it, to obtain the factor $G^*_e(x_{e\cap [i+1]},y_{e\cap [i+1]},x_{e\backslash [i+1]})$; thus, the formula holds for $i+1$. After finishing this process, we have by (A.2) and (A.3)

$$\begin{aligned} E^{2^d}\lesssim & {} \mathbb {E}_{x_{[d]},y_{[d]}} \prod _{\omega \in \{0,1\}^d} F_{e_0} (\omega (x_{[d]},y_{[d]}))\\&\times \, \prod _{f\subseteq [d],f\ne e_0}\prod _{\omega _f\in \{0,1\}^f} \nu _f(\omega _f(x_f,y_f))\,\mathcal {W}(x_{[d]},y_{[d]}), \end{aligned}$$

where

$$\begin{aligned} \mathcal {W}(x_{[d]},y_{[d]})=\mathbb {E}_{x_{d+1}} \prod _{d+1\in e\in \mathcal {H}} \prod _{\omega _e\in \{0,1\}^{e\cap [d]}} \nu _e (\omega _e (x_{e\cap [d]},y_{e\cap [d]},x_{e\backslash [d]})). \end{aligned}$$

Thus, as $F_{e_0}\le \nu _{e_0}$, to prove (1.26), it is enough to show that

$$\begin{aligned} \mathbb {E}_{x_{[d]},y_{[d]}} \prod _{f\subseteq [d]}\prod _{\omega _f\in \{0,1\}^f} \nu _f(\omega _f(x_f,y_f))\,|\mathcal {W}(x_{[d]},y_{[d]})-1|=o_{N \rightarrow \infty }(1). \end{aligned}$$

This, similarly as in [7], can be done with one more application of the Cauchy–Schwartz inequality leading to four terms involving the “big” weight functions $\mathcal {W}$ and $\mathcal {W}^2$. Each term is, however, $1+o_{N \rightarrow \infty }(1)$ by the linear form condition, as the underlying linear forms are pairwise linearly independent. Indeed, the forms $L_f(\omega _f(x_f,y_f))$ are pairwise linearly independent for $f\subseteq [d]$, and depend on a different set of variables rather than the forms $L_e(\omega _e (x_{e\cap [d]},y_{e\cap [d]},x_{e\backslash [d]}))$ for $e\nsubseteq [d]$ defining the weight function $\mathcal {W}$. The new forms appearing in $\mathcal {W}^2$ are copies of the forms in $\mathcal {W}$ with the $x_{d+1}$ variable replaced by a new variable $y_{d+1}$ hence are independent of each other and the rest of the forms. This proves the proposition. $\square$