13
votes
1answer
490 views

Is there a 'far' irrational number?

I recently learned of so-called 'far' numbers at a talk. In the talk, it was proven that there is a dense subset of the interval $[0,1]$ of far numbers (however, far numbers were only a minor point of ...
5
votes
1answer
170 views

Estimating $\#\{\{\alpha k\} < 1/\sqrt{k} : k \leq n\}$ for irrational $\alpha$

Suppose $\{\alpha n\}$ is the fractional part of $\alpha n$. Put $$A_{\alpha}(n) = \#\{\{\alpha k\} < 1/\sqrt{k} : k \leq n\}.$$ If $\alpha$ is irrational, can I find some constant $K$ such that ...