Product of circles and density of multiples of real numbers

Question

Is it true, that $$\left\{(\{r_1 k\} , \{r_2 k\} ) | k\in\mathbb{N})\right\}$$ (where the inner brackets denote the fractional part of a real number)
is dense in $S_1 \times S_1$, with appropriately weak assumptions on $r_1, r_2$, such as f.i. $r_1, r_2, \frac{r_1}{r_2} \notin \mathbb{Q}$?