most recent 30 from http://mathoverflow.net 2013-05-24T03:15:51Z http://mathoverflow.net/feeds/question/72721 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72721/diophantine-approximation unknown (yahoo) 2011-08-11T22:04:14Z 2012-09-05T19:25:28Z

<p>Say absolute values of $a,b,c$ is $O(log^{k}{n})$ for some positive constant $k$. Given positive integer $n$ that is reasonably large, we cannot always find integers $a,b,c$ such that $|a{b^{c}} - n|$ is very close to $n$ say within $O(log(n))$ since there are only $log^{O(k)}{n}$ such $a,b,c$ combinations(answer from Petrov). </p> <p>What is the lower bound on $n$ as a function of $k$ upto which I can find such $a,b,c$?</p>