Is the infimum of the product of the coordinates of a lattice positive?

Question

Let $L \subset \mathbb R^2$ be a lattice of rank $2$. Consider the set $$A := \{ x_1x_2 : x \in L \}.$$ Is the infimum of $A \cap (0,\infty)$ always positive?

Answer 1

I just worked it out myself. The answer is: The infimum is not always positive.

We give an example: Let $\alpha \in \mathbb R$ be an irrational number with approximation exponent $\sigma>2$. This means that the inequality $$\left|\alpha - \frac{p}{q}\right| < q^{-\sigma}$$ has infinitely many solutions in $\mathbb Z$. Without loss of generality we even have $$0<\alpha - \frac{p}{q}< q^{-\sigma}$$ for arbitrary large $q$.

We define the lattice $$L := \left\{\binom{\alpha q-p}{q} : p,q \in \mathbb Z \right\}.$$ Then we have $$ 0 < (\alpha q-p)q = q^2 \left( \alpha - \frac{p}{q} \right) < q^{2-\sigma}$$ for arbitrary large $q$. So the product of the coordinates gets arbitrary small.