Does a sequence of fractional parts of square root of two multiplied by n form a dense set on (0,1)?

Question

Is it true that the set of elements $mod(n\sqrt{2} , 1)$ where $n\in \Bbb{N}$ is dense in $(0,1)$? Is it true that for any $0<x<1$ and $\epsilon>0$ $(x-\epsilon, x+\epsilon)$ contains points $p$ from satisfying $p=fractional\_part(2^{1/2}*n)$?

Answer 1

Yes it is! You can think of your set as the orbit of a single point on the circle under a repeated irrational rotation, which reduces it to this question.