This question is an extension of my earlier question [here](https://mathoverflow.net/questions/448360/upper-bound-for-halls-conjecture-on-separation-of-squares-and-cubes), answered by Noam Elkies.


Let $A,B \in \mathbb{Z}$. Consider the inequality


$$\displaystyle |y^2 - x^3 - Ax - B| = O(|x|^{1/2 + \theta}).$$


For general $A,B$, what is the smallest $\theta \geq 0$ (independent of $A,B$) for which the above inequality has infinitely many solutions in integers $x,y$?


The construction I gave in the question linked above still works in the general case, which shows that $\theta = 1/2$ is admissible. The polynomial identity which works for the case $A = B = 0$ does not seem easy to generalize to the general case.



Integral points near elliptic curves
You can take $\theta = 0$, even $\theta = -1/6$ works.


Fix an integer $A \ne 0$ and an integer $B$. If $r$ is an integer, the elliptic curve $E : y^{2} = x^{3} + Ax + r^{2} A^{2}$ has the obvious point $P = (0,rA)$. Somewhat miraculously, the point $3P$ is integral. Setting $y = 512 A^{3} r^{9} + 96A^{2} r^{5} + 3Ar$ and $x = 64 A^{2} r^{6} + 8Ar^{2}$ gives
$$
 |y^{2} - x^{3} - Ax - B| = |A^{2} r^{2} - B| = O(x^{1/3})
$$
if $r$ is large.


