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The affine sieve, developed initially by Bourgain-Gamburd-Sarnak in the paper "Affine linear sieve, expanders, and sum-product" published in Inventiones Mathematicae in 2010, deals generally with the following problem: given a subgroup $\Lambda \subset \text{GL}_n$ and a point $b \in \text{GL}_n$, let $\mathcal{O} = \mathcal{O}_b = b \cdot \Lambda$ be the orbit of $b$ under $\Lambda$. If $f \in \mathbb{Z}[x_{jk}]$ satisfies the condition that for all $q \geq 1$ there exists $x \in \mathcal{O}$ such that $\gcd(f(x), q) = 1$m then we say that $f$ is primitive. Let $r_0(\mathcal{O}, f)$ be the least positive integer $r$ such that the set $\{x \in \mathcal{O} : f(x) \text{ has at most } r \text{ prime factors}\}$ is Zariski dense in the Zariski closure of $\mathcal{O}$. If no such integer exists, then we say $r_0(\mathcal{O}, f) = \infty$.

I want to ask if it is possible to use the affine sieve to prove a result of the following shape: Define $k_0(\mathcal{O}, f)$ to be the least positive integer $k \geq 2$ such that the set $$\displaystyle \{x \in \mathcal{O} : f(x) \text{ is } k\text{-free}\}$$ is Zariski dense in the Zariski closure of $\mathcal{O}$. An integer $n$ is said to be $k$-free if for all primes $p | n$, we have $p^k \nmid n$. The $k=2$ case gives the familiar square-free numbers. For a given matrix $\gamma \in \text{GL}_n$, write $\lVert \gamma \rVert$ for the Hilbert-Schmidt norm of $\gamma$. Can the affine sieve be used to prove a lower bound, or even an asymptotic formula, for the quantity $N_{f, \mathcal{O}}(T) = \# \{x \in \mathcal{O} : f(x) \text{ is } k\text{-free}, \lVert x \rVert \leq T\}$?

Initially I thought an obstruction was that a crucial part of the proof of the main theorem of the affine sieve requires showing that the family of Cayley graphs $\mathcal{G}(\Lambda/\Lambda(q), S)$ forms an expander family (here $S$ is a finite symmetric set which generates $\Lambda$) as $q$ ranges over the square-free numbers. The requirement for $q$ to be square-free seems to be the main obstacle, but the above theorem was proved for the case when $q$ ranges over all positive integers in a 2012 paper by Bourgain and Varju. However, I have not seen this information being applied to improve the main theorem of the affine sieve in my literature review.

Any information would be appreciated.

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Given that $\sum_q \frac{1}{q^k}$ converges, I would imagine that one could get asymptotic formulae here from much simpler methods than the affine sieve (e.g. strong approximation property + Lang-Weil); basically one just needs an asymptotic for $\# \{ x \in {\mathcal O}: q^k | f(x), |x_i| \leq B_i \}$ for each fixed squarefree $q$ with a main term that is summable with $q$, with no uniformity required on the error term (as long as one also has an upper bound comparable to the main term). But the affine sieve may help get better error terms than what an elementary method would give. –  Terry Tao Jun 10 at 16:33
    
Edited it to reflect closer to the spirit of the affine sieve, which is to look at those $f$ that takes on values in the coordinates of the underlying matrix group. –  Stanley Yao Xiao Jun 11 at 20:19

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