Let $g : \mathbb{N} \rightarrow \mathbb{N}$ be an increasing function, and consider the sequence $Y = \{y_n\}$ given by $y_n = g(n)$. Let $F$ be an irreducible binary form of degree $d \geq 2$ with integer coefficients such that $F$ does not vanish identically modulo $p$ for any prime $p$. We say that the sequence $Y$ captures its primes with respect to $F$ if there exists a real function $u(x)$ tending to infinity such that the number $N(Y,F,x)$ of primes $p \leq x$ for which there exists an integer $x$ and a natural number $n$ satisfying $p = F(x,y_n)$ satisfies the asymptotic relation
$$\displaystyle N(Y,F,x) \sim u(x).$$
It is a classical result that the sequence of positive integers captures its primes with respect to any admissible, irreducible binary quadratic form; the most classical case where $F = F_0(x,y) = x^2 + y^2$ being claimed by Fermat and ultimately proved by Euler. A landmark paper of Friedlander and Iwaniec showed that the sequence of squares captures its primes with respect to $F_0$ ( J.B. Friedlander and H. Iwaniec, The polynomial $X^2 +Y^4$ captures its primes, Ann. of Math. 148 (1998), 945–1040). Heath-Brown proved that the sequence of positive integers captures its primes with respect to the cubic form $F_1 = x^3 + 2y^3$. Fouvry and Iwaniec showed that the sequence of primes captures its primes with respect to $F_0$, and very recently, Heath-Brown and Li showed that the sequence of prime squares captures its primes with respect to $F_0$ (http://link.springer.com/article/10.1007/s00222-016-0694-0).
In particular, the Heath-Brown/Li result shows that a sequence as thin as $o(x^{1/2})$ captures its primes with respect to $F_0$.
My question is an inverse of these theorems, namely to ask what conditions $Y$ must satisfy in order for it to be reasonable for it to capture its primes with respect to some binary form $F$, and more importantly, what conditions should $Y$ satisfy in order for one to be able to prove that it captures its primes with respect to a binary form $F$. We can restrict to the case when it is known that the sequence of positive integers captures its primes with respect to $F$; and we know this for $d = 2,3$.
