Question

Let $x$ be a real number and $N$ a positive integer. Define

$E(N,\delta) = \{(p,q) \in \mathbb{Z}^2: |p - q x| \leq \frac{\delta}{N}, |p|, |q| \leq N \}$,

i.e., the set of solutions to rational approximation of $x$ with accuracy $\frac{\delta}{N}$.

I am interested in the behavior of the cardinality of $E(N,\delta)$. Question:

For which $x$ do we have $|E(N,\delta)| \leq c(\delta) N$ where $c(\delta) \to 0$ as $\delta \to 0$?

Of course $x$ has to be irrational. Is this true for all irrational $x$? I am very unfamiliar with Diophantine approximation. I googled a bit and found that Schmidt proved that $|E(N,\delta)| = O(\log N)$ for a.e. $x$. Lang proved that this holds for all quadratic irrational $x$. But $O(\log N)$ is much stronger than what I asked, which is even weaker than $o(N)$.

(One further question: if we replace $\frac{1}{N}$ by $\frac{1}{N^{1+\epsilon}}$, how does the number of solutions behave?)

Answer 1

It's usually written $p-qx$. If $\delta\lt1/2$, then $|(p/q)-x|\le(1/2)q^{-2}$, which, if I remember right, implies, by a theorem of Hurwitz, that $p/q$ is a convergent to the continued fraction of $x$. It should not be hard to show that the number of convergents to $x$ with denominator not exceeding $N$ is little-oh of $N$.