What numbers can be approximated "pretty well" by rationals? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T09:39:09Z http://mathoverflow.net/feeds/question/43381 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43381/what-numbers-can-be-approximated-pretty-well-by-rationals What numbers can be approximated "pretty well" by rationals? Qiaochu Yuan 2010-10-24T14:05:58Z 2010-10-26T19:35:50Z <p>More precisely, what real numbers $r$ have the following property: for any $\epsilon > 0$ there exist infinitely many pairs $(p, q)$ of integers such that</p> <p>$$\left| \frac{p}{q} - r \right| &lt; \frac{\epsilon}{q^2}.$$</p> <p>I think that this is impossible if $r$ is a quadratic irrational. On the other hand, it's certainly possible for any number with <a href="http://mathworld.wolfram.com/IrrationalityMeasure.html" rel="nofollow">irrationality measure</a> strictly greater than $2$.</p> <p>What I really want to know is if the real numbers which don't have the property above have measure zero. If that's true, it would answer <a href="http://math.stackexchange.com/questions/7634/you-are-standing-at-the-origin-of-an-infinite-forest-holding-an-infinite-bb-gu/" rel="nofollow">the last part of this math.SE question</a>. </p> http://mathoverflow.net/questions/43381/what-numbers-can-be-approximated-pretty-well-by-rationals/43385#43385 Answer by Felipe Voloch for What numbers can be approximated "pretty well" by rationals? Felipe Voloch 2010-10-24T14:30:48Z 2010-10-24T14:30:48Z <p>This is a well-studied question in diophantine approximation. You can look up Markov or Lagrange spectrum for a "description" of the numbers for which you cannot take $\epsilon$ arbitrarily small. For the answer to your last question, look up Khinchin's theorem (the answer is no, they have full measure).</p> http://mathoverflow.net/questions/43381/what-numbers-can-be-approximated-pretty-well-by-rationals/43560#43560 Answer by Hany for What numbers can be approximated "pretty well" by rationals? Hany 2010-10-25T18:53:36Z 2010-10-26T19:35:50Z <p><a href="http://books.google.com.eg/books?id=rey9wfSaJ9EC&amp;printsec=frontcover&amp;dq=Hardyand+wright+number&amp;source=bl&amp;ots=avj7HPEGSa&amp;sig=ZBqVYG6xeXOSbwcC5mUxM-ql_uQ&amp;hl=ar&amp;ei=pdHFTLeQOca54gb28OC6Aw&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=10&amp;ved=0CEgQ6AEwCQ#v=onepage&amp;q&amp;f=false" rel="nofollow">Hardy and Wright</a> devoted a chapter (chapter 9) to these questions. One interesting theorem related to your question is theorem 196.</p>