Hall's (weak) conjecture is the statement that for all $\varepsilon > 0$ there exists a positive number $c(\varepsilon) > 0$ such that for all $x,y \in \mathbb{Z}$ with $y^2 \ne x^3$, that

$$\displaystyle \left \lvert y^2 - x^3 \right \rvert \geq c(\varepsilon) |x|^{\frac{1}{2} - \varepsilon}.$$

My question concerns a related question. What is the smallest $\theta \geq 0$ for which it is known that there exist infinitely many $x,y \in \mathbb{Z}$ with $y^2 \ne x^3$ satisfying

$$\displaystyle |y^2 - x^3| \ll |x|^{\frac{1}{2} + \theta}?$$

The choice $\theta = 1/2$ can be achieved as follows: write $x^3 = (x-1)^2(x+2) + 3x - 2$, and choose $x = t^2 - 2$ and $y = (t^2 - 3)t$. Then

$$y^2 - x^3 = (t^2 - 3)^2t^2 - (t^2 - 3)^2 t^2 + 3(t^2 - 2) - 2 = 3t^2 - 8,$$

which is manifestly $O(t^2) = O(|x|)$. Here we can freely choose $t \in \mathbb{Z}$, so this gives infinitely many integers satisfying the required inequality.

Can one do better than $\theta = 1/2$?