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Green-Tao-Ziegler theorem implies that for the localized integers

In a paper on rational points on varieties, they exploit the Green-Tao-Ziegler theorem. One way to state the theorem is as follows.

Let $L_i(x,y)\in \mathbb Z [x,y]$ (i=1, ..., r) be finitely many homogeneous polynomials of degree 1 and assume none is a rational multiple of another. Let $c_i\in \mathbb Z$ be integers and $K\subset \mathbb R^2$ a cone (recall that a cone is a subset stable under multiplication by positive real numbers).

Assume that for every prime $p$, the map $\mathbb F_p^2\to \mathbb F_p$ defined by the polynomial $\prod  _{i=1}^r ( L_i(x,y)+c_i )$ is not the zero function.

Then there exist infinitely many $(m,n)\in K\cap \mathbb Z^2$ such that $L_i(m,n)+c_i$ are all prime numbers.

The actual theorem describes the asymptotic number of such pairs $(m,n)$ but let's not focus on it right now.

The $\mathbb Z _S$ version

It might be useful to have the same statement when finitely many primes are inverted:

Let $S$ be a finite set of primes. Let $\mathbb Z _S=\mathbb Z[S^{-1}]$ be the ring of integers with the primes in $S$ inverted.

In parallel with the above theorem, suppose $L_i(x,y)\in \mathbb Z_S[x,y]$, $c_i\in \mathbb Z_S$ and the assumption is only considered for $p\not\in S$ (that is, for $(p)\in Spec \mathbb Z _S$). Then there exist $(m,n)\in K\cap \mathbb Z^2$ arbitrarily far away from the origin, such that $L_i(m,n)+c_i$ are all prime elements of $\mathbb Z_S$.

In this post we show how the second statement can be deduced from the first.

In an earlier version of this post I wrote there exist many $(m,n) \in K\cap \mathbb Z _S^2$ but now I state more strongly $(m,n)\in K\cap \mathbb Z^2 $ partly because the argument below actually shows this and also because this seems to have real advantages.

SO THE FOLLOWING PARAGRAPH IS NOW OBSOLETE: Let me remark that I have phrased the statement by saying (m,n) should be "arbitrarily far away from the origin" rather than "infinitely many." This is because $\mathbb Z_S$ is dense in $\mathbb R$ as long as $S$ is non-empty and therefore there are infinitely many elements of bounded size. But I guess one wants to have (m,n) having large values.


The arguments are simple and taken from part of the proof of Proposition 1.2 of the paper.

Let $q_S\in \mathbb Z$ denote a number obtained as a product of high powers of $p\in S$. Consider degree 1 polynomials of the form \[ (\text{an element of }\mathbb Z_S^*)\cdot ( L_i(q_S x,q_S y)+c_i ) . \quad (*) \] If $q_S$ is large enough, we see that the coefficients of  $L_i(q_Sx,q_Sy)$ = $q_S L_i (x,y)$ are in $\mathbb Z$ and contain higher powers of $p\in S$ than $c_i$ does. Then we choose an appropriate element of $\mathbb Z_S^*$ so that the constant term becomes an element of $\mathbb Z$ coprime to all $p\in S$.

Thus we have obtained degree 1 polynomials with $\mathbb Z$-coefficients. We can ask if they satisfy the assumption in the Green-Tao-Ziegler theorem. Namely we ask if $\mathbb Z_S^*\cdot \prod _i ( L_i(q_S x,q_S y)+c_i )$ is a zero function $\mathbb F_p^2\to \mathbb F_p$ or not for primes $p$.

We use different arguments according to whether $p\in S$ or not. When $p\not\in S$, recall that we are already assuming something similar to the assumption in Green-Tao-Ziegler. Since elements in $\mathbb Z_S^*$ and $q_S$ are all invertible in $\mathbb F_p$, one can readily verify the assumption in Green-Tao-Ziegler for the polynomials $(*)$.

Suppose $p\in S$. If we set $x=y=0$, the value is of the form $\mathbb Z_S^*\cdot c_i$. Recall that the $\mathbb Z_S^*$ part has been chosen so that the product is coprime to $p$. One concludes that this case is also fine.


Thus we find infinitely many $(m,n)\in \mathbb Z^2\cap K$ for which $\mathbb Z_S^*\cdot (L_i(q_S m,q_S n) +c_i)$ are simultaneously prime numbers. This is equivalent to that $L_i(q_S m,q_S n )+c_i$ is a prime element of $\mathbb Z_S$. The pair $(q_S m,q_S n)$ is in the cone $K$ and can be arbitrarily far from the origin. This completes the proof. ◼️



One would naturally wonder what the number field analog of Green-Tao-Ziegler should be and if the Green-Tao-Ziegler for $\mathcal O_F$ implies that for localized $\mathcal O_F$. One can formulate Green-Tao-Ziegler for $\mathcal O_{F,S}$ simply by replacing every $\mathbb Z_S$ by $\mathcal O_{F,S}$. I don't know if this is a reasonable statement to expect. I hope it is.



Green-Tao-Ziegler for $\mathcal O_{K,S}$ implies Prop.1.2 for $\mathcal O_{K,S}$ 

It is routine to check Prop.1.2 if we focus only on the condition that $\lambda -e_i \mu $ be prime elements.

To guarantee the condition $| \phi ( \lambda /\mu ) | > c$ for all embedding $\phi \colon K\to \mathbb C $ (and that $\phi (\lambda /\mu ) $ should furthermore be positive if $\phi $ is real) and the condition that $\lambda - e_i \mu $ determine distinct prime ideals, we have to take an appropriate open convex cone $C\subset (K\otimes _{\mathbb Q} \mathbb R )^2$.

(Here, the right notion of a cone in $K\otimes _{\mathbb Q}\mathbb R \cong \mathbb R^{r_1} \times \mathbb C^{r_2}$ might be a subset stable under the multiplication by $\mathbb R_{>0}$ or one stable under the multiplication by $\mathbb R _{>0}^{r_1}\times \mathbb C^{r_2}$. This choice would depend on the application one has in mind.)

Note that:

Any open cone in $\mathbb R^n $ contains an open convex cone. (At least if cone is interpreted as a subset stable by $\mathbb R_{>0}$.)

So we only have to find an open cone (not necessarily convex) in which the requirements are satisfied. One might wonder then why Green-Tao-Ziegler phrased their theorem using open convex cones. It's because their statement contains an asymptotic formula for whose proof they use the convexity.

Recall that the first requiment is $| \phi (\lambda /\mu ) | > c $ for all embeddings $\phi $. Recall the way we are taking $(\lambda ,\mu )$: \[  (\lambda ,\mu )= (\lambda _0 ,\mu _0 ) + M\cdot (m,n) . \] Let $c'>c>0$ be any number larger than $c$. If $|\phi ( m/n )| >c' $ and if $| \phi (m)|$ and $|\phi (n)|$ are large enough depending on $\lambda _0,\mu_0 ,c' $ and $M $ then we will have $|\phi (\lambda /\mu ) | >c $. 

So let us restrict ourselves to those (m,n) with: \[  | \phi (m/n) | > c' . \] 

Making them different as prime ideals

Next requiment was that $\lambda -e_i \mu $ should determine distinct prime ideals of $\mathcal O_{K,S} $. We will achieve this by ensuring that the cardinalities \[ |\mathcal O_{K,S} / (\lambda -e_i \mu ) |  \] are different.

Recall the product formula: for any $\alpha \in K^*$ we have \[ \prod _{v: \text{ all places} } |\alpha |_v =1,  \] where for safety we recall $|\alpha |_{\mathfrak p} =  |\mathcal O_{K,S}/\mathfrak p |  ^{- v_{\mathfrak p} (\alpha )}$, $|\alpha |_v = |\phi (\alpha ) |$ if $v$ corresponds to $\phi \colon K\to \mathbb R$ and $|\alpha |_v = |\phi (\alpha ) |^2$ if $v$ corresponds to an imaginary $\phi \colon K\to \mathbb C $.

By prime ideal decomposition we know:  \[ |\mathcal O_{K,S} / (\lambda -e_i \mu ) | =\prod _{\mathfrak p\in Spec \mathcal O_{K,S} } |\mathcal O_{K,S} /\mathfrak p | ^{v_{\mathfrak p } (\lambda -e_i \mu ) }  . \] Since finite places are either $\in Spec \mathcal O_{K,S}$ or $\in S$, Product Formula implies that this equals: \[ = \prod _{v \in S } |\lambda - e_i \mu |_{v} \cdot \prod _{v|\infty } |\lambda -e_i\mu |_v . \] The values $|\lambda -e_i \mu |_{\mathfrak p}$ for $\mathfrak p\in S $ are determined already by the approximation condition; since $(\lambda , \mu )$ is required to be close to the given $(\lambda _{\mathfrak p},\mu _{\mathfrak p} )$, we have \[ |\lambda -e_i \mu |_{\mathfrak p} = |\lambda _{\mathfrak p} -e_i \mu _{\mathfrak p} |_{\mathfrak p} =: N_i . \] Therefore the condition that $|\mathcal O_{K,S} /(\lambda -e_i\mu ) |$ be different from each other is equivalent to the condition: \[\prod _{v|\infty } \frac{ |\lambda -e_i\mu |_v}{|\lambda -e_j\mu |_v} \neq  \frac{N_j}{N_i} \quad\text{ for all }i\neq j . \] 

How to choose the cone

Several formulas below are to be interpreted in $\mathbb C$ via a chosen embedding $\phi \colon K\to \mathbb C$ (real or imaginary). Note that the complex number \[ \frac{ \lambda -e_i\mu }{ \lambda -e_j\mu } = \frac{ (\lambda _0 -e_i\mu _0 )+M (m-e_i n) }{ (\lambda _0 -e_j\mu _0 )+M (m-e_j n) }  \] is very close to \[ \frac{m-e_i n}{m-e_j n} = \frac{(m/n) - e_i}{(m/n) -e_j } \] when $\phi (m)$ and $\phi (n)$ have large absolute values. Recall that the correspondence $\theta \mapsto \frac{\theta -e_i}{\theta -e_j }$ gives an automorphism of $\mathbb{P}^1$ (because $e_i\neq e_j $). So the displayed ratio can be arbitrarily close to any chosen real or complex number by setting $m $ and $n$ large and $m/n$ close to an appropriate complex number. 

Fix your favorite $\phi _0\colon K\to \mathbb C $ (real or complex). Since there are only finitely many pairs of indices $i\neq j$, for a generic choce of $\theta _0 \in \mathbb C$ (or $\in \mathbb R$ if $\phi _0$ is real) we have \[ \left| \frac{\theta _0 -e_i}{\theta _0 -e_j } \right| \neq \frac{N_j}{N_i} \quad\text{ for all }i\neq j . \] Fix such a $\theta _0$ whose absolute value is larger than $c'$ (see the previous paragraph for $c'$). Consider the cone $C\subset (K\otimes _{\mathbb Q} \mathbb R )^2 = \{ (x_\phi , y_\phi )_{\phi \colon K\to \mathbb C } \} $ defined by: \[ C: \begin{array}{rl}  & x_{\phi _0} /y_{\phi _0}  \text{ is close enough to $\theta _0$} \\ & x_\phi /y_\phi \text{ is large enough (in particular $>c'$) for $\phi \neq \phi _0 $ so that $\frac{ (x_{\phi } /y_{\phi } )-e_i }{ (x_{\phi } /y_{\phi } )-e_j } \approx 1 $}   , \end{array} \] where the precision needed for "close enough" and "large enough" is determined so that the approximations "$\approx $" are close enough to obtain the final conclusion.

When $(m,n) \in K^2$ is in this cone and far enough from the origin (in that $m $ and $n$ has large absolute values at every infinite places) we have \[ \prod _{\phi \colon K\to \mathbb C} \frac{\phi (m/n) -e_i }{\phi (m/n )-e_j } \approx \frac{\theta _0 -e_i}{\theta _0 -e_j } \cdot 1\cdot \dots \cdot 1 \] which is $\neq \frac{N_j}{N_i}$ for any $i\neq j$. It follows that for such $(m,n)$ and the corresponding $(\lambda ,\mu )$, the cardinalities $|\mathcal O_{K,S}/(\lambda -e_i \mu )| $ are different for different $i$. In particular $(\lambda -e_i \mu )$ are different ideals. ◼️