(This post follows an email by Aicke Hinrichs.)
In a previous post we discussed the following problem:
Problem: Let be a measurable subset of the -dimensional sphere . Suppose that does not contain two orthogonal vectors. How large can the -dimensional volume of be?
Setting the volume of the sphere to be 1, the Frankl-Wilson theorem gives a lower bound (for large ) of ,
2) The double cap conjecture would give a lower bound (for large ) of .
A result of A. M. Raigorodskii from 1999 gives a better bound of . (This has led to an improvement concerning the dimensions where a counterexample for Borsuk’s conjecture exists; we will come back to that.) Raigorodskii’s method supports the hope that by considering clever configurations of points instead of just -vectors and applying the polynomial method (the method of proof we described for the Frankl-Wilson theorem) we may get closer to and perhaps even prove the double-cap conjecture.
What Raigorodskii did was to prove a Frankl-Wilson type result for vectors with coordinates with a prescribed number of zeros. Here is the paper.
Now, how can we beat the record???