More than \(1-\frac{1}{n}\) of the circumference of a circle is colored black. Can you draw a regular \(n\)-gon on the circle with all vertices in black regions?
this is yet again problem from the book.
- solve assuming the black area is in finitely many disjoint intervals
- solve assuming the axiom of choice and not assuming anything beyond the measure of the set of black points being larger than \(1-\frac{1}{n}\).
um. I’m going to pretend like I didn’t think up question 2. but I’m kind of curious. could it really be different? anyways, here’s my solve for question 1:
Proof. Imagine a discrete version of the problem. It is trivial.
Now, if we discrete the original problem correctly we get this trivial discretized version.