I learned about these from professors and grad students: to my knowledge, they’re all still open. I’m happy to talk about any one of these!
Prove or disprove the following:
- The sequence 2^n sqrt(2) mod 1 is dense in [0,1]
- If f : R^n to R is smooth, and strictly, locally minimized at the origin along any smooth arc through the origin, then f is strictly, locally minimized at the origin. (note, this is NOT true if we replace “smooth arc” with “straight line”.)
- Talagrand’s convexity problem: my idea is to try to find a quantitative Steinhaus’ theorem, which might suggest whether to prove or disprove the claim. Problem is stated in a downloadable .pdf here: https://michel.talagrand.net/prizes/convexity.pdf