number of integers $n$ with $\|n \alpha \|$ small?

Question

Let $\alpha \in \mathbb{R}$ and $N$ a positive integer. I am interested in the quantity $$ D(\alpha, N) := \# \{ n \in [1, N]: \| n \alpha \| < 1/N \}, $$ $\| x \|$ denotes the distance to the closest integer. Dirichlet's approximation theorem implies $D(\alpha, N) \geq 1$ for all $N$. What I am interested knowing is does there exist $\alpha$ such that $D(\alpha, N) \leq C$ for all $N \geq 1$? Any reference is appreciated! Thank you!

Answer 1

Yes. Assume that $\alpha$ is irrational, and its continued fraction digits do not exceed $K$. Then, for any positive integer $q$, we have $$q\|q\alpha\|>1/(K+2).$$ In particular, for $q\leq N$, we have $\|q\alpha\|>1/(N(K+2))$. This implies that $D(\alpha,N)\leq 2K+4$, because one cannot accommodate $2K+5$ real numbers in $(-1/N,1/N)$ with pairwise distances exceeding $1/(N(K+2))$.