Let $a,b,c$ be co-prime positive integers and let $\theta \in (1/4, 1/3)$ be a real number. For each positive integer $k$, does there exist a positive integer $N$ such that the linear diophantine equation

$$\displaystyle ax - by = c$$

has at least $k$ solutions with $x, y$ dividing $N$, and $N^\theta \leq x,y \leq 2N^\theta$?

Without the bound on the solutions, the answer is clearly yes. Indeed, simply take any solution $(x_0, y_0)$ of the LDE and generate a sequence $(x_0 + jb, y_0 + ja)$ for $j = 1, \cdots, k$ and let

$$\displaystyle N = x_0 y_0(x_0 + b)(y_0 + a) \cdots (x_0 + kb)(y_0 + ka).$$

However, this construction doesn't work with the size restrictions, since the "obvious" solutions above will eventually be too small for $k$ large.

This is a special instance of an $S$-unit equation, known to always have finitely many solutions but generally not uniformly bounded, but this is far more restrictive since the solutions have to be rational integers, and the exponents have to be bounded.