Consider the inequalities


$$\frac{(2A-1)^2}{4A^2}xy\leq \Big(\frac{x+y}2\Big)^2\leq\frac{(2A-1)^2}{4(A-1)^2}xy$$
$$x,y\geq0$$ where $A>10^9$.


Is the set of integer solutions to $x,y$ finite?



Is set of integer solutions to these inequalities finite?
No. If you consider the inequalities in $\mathbb R^2$, the set of solutions is invariant under scaling and contains an open set. Hence it contains arbitrarily large balls and so infinitely many integer points.


