Science Talk

Yihong · Guest

Hi all,

I have a question on Steinhaus theorem which states as follows,

For any Borel set A in R^n with nonzero Lebesgue measure, put A-A = {x - y: x, y \in A}. Then there exists delta > 0, s.t. the open ball centered at 0 with radius delta is contained in A-A.

My question is whether there is any generalization on this theorem, in the sense that the positivity of Lebesgue measure is _not_ necessary. For example, let C be the ternary Cantor set in the unit interval. Then it can be shown that C-C = unit interval, hence containing an open ball.

Thanks a lot!

Best,
Yihong