Proving the Existence of Rational Differences in a Measurable Set

[modestoraton]

If i have a measurable set with positive measure, how do I prove that there are 2 elements who's difference is in Q~{0} (aka a rational number that isn't 0.

 

[micromass]

Hi modestoraton!

Let M is measurable such that there are no two elements who's differences are in \mathbb{Q}\setminus \{0\}. Let M_n=M\cap[n,n+1].

Then perhaps you could evaluate the sum

\lambda\left(\bigcup_{q\in\mathbb{Q}\cap[-1,1]}{q+M_n}\right)

and show that M_n has measure zero.

 

[modestoraton]

Thank you so much.