I have that $f(x)$ and $g(x,y)$ are real polynomial with no constant terms. We have that $\forall \epsilon > 0$ the set
$S_{\epsilon}$ = { $x \in\mathbb{Z} :$ Frac(f(x)) $ < \epsilon $ } is infinite, where Frac(.) is the fractional part function.
Question is whether $T_{\epsilon}$ = { $ (x,y)\in\mathbb{Z}^{2} : $Frac(g(x,y)) $< \epsilon $ } is infinite $\forall \epsilon > 0$ ?
I tried working with examples : $f(x) = \sqrt{2} x $ and $g(x,y) = \pi xy $ but I couldn't achieve anything. Can anyone help me with this problem, or even the example that I have selected.
Note : When all the coefficients of the polynomials are rational, this problem is trivial. Therefore I am taking coefficients $\sqrt{2}$ and $\pi$.