Trying to prove that a region contains infinitely many lattice points

Question

Let $S$ be a region in $\mathbb{R}^2$ defined to be $$ S = \{ (x, y) \in \mathbb{R}^2: \lambda_1 (x^2 + y^2)^{c} < ax + by < \lambda_2 (x^2 + y^2)^{c} \}, $$ where $\lambda_1 < \lambda_2$ and $c > 0$ and $(a, b) \neq (0,0)$. I am trying to prove that $S$ contains infinite number of points with integer coordinates $(x,y) \in \mathbb{Z}^2$. Since the difference in inequality goes to infinity, I think it's definitely true, but I couldn't come up with a simple argument to prove it... Does anyone see a simple way to prove it? Thank you...

Answer 1

The statement is not true in general. For a concrete example, let $a=1, b=0, c=1, \lambda_1=1$. Then the lower inequality becomes $x^2+y^2<x$, which can be written as $$ y^2<x(1-x) $$ which implies that $x(1-x)$ is positive, hence $x$ and $1-x$ both have the same sign. Since they add to a positive number, both must be positive and thus $0<x<1$, which means there are no integer solutions to this inequality.