Approximating multiples of reals with integers

Mathematics Asked on January 25, 2021

I have recently recalled a problem from my school days, where I was asked if for every $varepsilon > 0$ there exist positive integers $m, n$ such that $|msqrt 2 – n|< varepsilon$. I have just realized, that I do not really know of a simple way to solve this problem. Furthermore, would be good if that solution could be applied also to any $qin Bbb R_+$ instead of $sqrt2$.

real analysis

One Answer

Let $0<qinBbb RsetminusBbb Q$, $M={mq-n:m,ninBbb Z}$ and $a=inf{x:xin Mland x>0}$. Assume on contrary $a>0$.

If $ain M$, then for every $0<xin M$ we have $0leq x-lfloor x/arfloor a<a$ and since $x-lfloor x/arfloor ain M$ this implies $x=lfloor x/arfloor ain aBbb Z$. Then $q=au$ and $1=av$ for some $u,vinBbb Z$, and we get the contradiction $q=u/vinBbb Q$.

If $anotin M$, by definition of infimum, then there exists $m',n'inBbb Z$ such that $a<m'q-n'lt 2a$ and, again by definition of infimum, there exists $m,ninBbb Z$ such that $a<mq-nlt m'q-n'$. Then $0lt(m'-m)q-(n'-n)lt a$ - a contradiction.

Answered by Fabio Lucchini on January 25, 2021

Add your own answers!

Ask a Question

Get help from others!