Affine Linear Sieve, Expanders, and Sum-product

Abstract

This paper is concerned with the following general problem. For j = 1, 2, . . . , k let Aj be invertible integer coefficient polynomial maps of Z to Z (here n ≥ 1 and the inverses of Aj’s are assumed to be of the same type). Let Λ be the group generated by A1, . . . , Ak and let O = Ob = b · Λ be the orbit of some b ∈ Z under Λ. Given a polynomial f ∈ Q[x1, . . . , xn] which is integral on O our aim is to show that there are many points x ∈ O at which f(x) has few or even the least possible number of prime factors, in particular that such points are Zariski dense in the Zariski closure, Zcl(O) of O. Let O(f, r) denote the set of x ∈ O for which f(x) has at most r prime factors. As r → ∞ the sets O(f, r) increase and potentially at some point become Zariski dense. Define the saturation number r0(O, f) to be the least integer r such that Zcl(O(f, r)) = Zcl(O). It is by no means obvious that that r0(O, f) is finite or even if one should expect it to be so in general. If it is finite we say that the pair (O, f) saturates. Many classical results and conjectures are concerned with this problem in the case that Λ is a subgroup of Z acting by translations, that is Aj(x) = x + bj. For example if Λ = qZ, O = b + Λ and f(x) = x one checks that Dirichlet’s Theorem [18] is equivalent to r0(O, f) = 1 + ν ( (b, q) ) , where ν(m) is the number of prime divisors of m. Another example is Λ = Z,O = Z and f(x) = x(x + 2). Brun [12] invented the combinatorial sieve to show that this pair (O, f) saturates; the twin prime conjecture is equivalent to r0(O, f) = 2. One can use the classical combinatorial sieve in Z along the lines of Section 3 below, to show that any pair (O, f) with Λ ⊂ Z acting by translations saturates. One of the main goals of this paper is to study the case that

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