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.
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Notes
The family \(\{\nu _e\}\) can be considered as a parametric family of weights in a trivial way, setting \(Z=\Omega =\{0\}\), and \(\psi (0)=1\).
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Ákos Magyar is supported by NSERC Grant 22R44824 and ERC-AdG. 321104.
Appendix A: Basic Properties of Weighted Box Norms
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
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
Applying the Cauchy–Schwartz inequality in the \(x_1\) variable, one has
here,
where
for any \(\omega _{[2,d]}\). Similarly
where
for any \(\omega _{[2,d]}.\) In the same way, applying the Cauchy–Schwartz inequality in \(x_2\) variable, we end up with
and continuing this way with \(x_3,\ldots ,x_d\) variables, we end up with
\(\square \)
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
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
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
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,
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
We claim that after applying the Cauchy–Schwartz inequality in the \(x_1,\ldots ,x_i\) variables, we have with \(g=[i]\)
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)
where
Thus, as \(F_{e_0}\le \nu _{e_0}\), to prove (1.26), it is enough to show that
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. \(\square \)
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Cook, B., Magyar, Á. & Titichetrakun, T. A Multidimensional Szemerédi Theorem in the Primes via Combinatorics. Ann. Comb. 22, 711–768 (2018). https://doi.org/10.1007/s00026-018-0402-4
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DOI: https://doi.org/10.1007/s00026-018-0402-4