number of solutions of diophantine approximation - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T18:20:13Z http://mathoverflow.net/feeds/question/51445 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51445/number-of-solutions-of-diophantine-approximation number of solutions of diophantine approximation mr.gondolier 2011-01-07T23:15:34Z 2011-01-08T00:50:53Z <p>Let $x$ be a real number and $N$ a positive integer. Define</p> <p>$E(N,\delta) = \{(p,q) \in \mathbb{Z}^2: |p - q x| \leq \frac{\delta}{N}, |p|, |q| \leq N \}$,</p> <p>i.e., the set of solutions to rational approximation of $x$ with accuracy $\frac{\delta}{N}$.</p> <p>I am interested in the behavior of the cardinality of $E(N,\delta)$. Question:</p> <p>For which $x$ do we have $|E(N,\delta)| \leq c(\delta) N$ where $c(\delta) \to 0$ as $\delta \to 0$?</p> <p>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 $o(N)$.</p> <p>(One further question: if we replace $\frac{1}{N}$ by $\frac{1}{N^{1+\epsilon}}$, how does the number of solutions behave?)</p> http://mathoverflow.net/questions/51445/number-of-solutions-of-diophantine-approximation/51450#51450 Answer by Gerry Myerson for number of solutions of diophantine approximation Gerry Myerson 2011-01-08T00:16:09Z 2011-01-08T00:16:09Z <p>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 $N$. </p>