A question by Furstenberg

Yesterday I was talking about some properties of different dimensions with Furstenburg. Somehow I mentioned Kekaya, and he told me about the following question he has been longing to solve (which is amazingly many similarities to Kekaya):

For set , if s.t. for all direction line with direction s.t. . Does it follow that ?

Note that a stronger conjecture would be is at least which when taking would give a generalization of the -dimensional Kekaya. (i.e. instead of having to have a line segment, we only require a 1-dimension set in each direction)

Reviewing the proofs of the 2-dimsional Kekaya, I found they all rely on the fact that the line segment is connected…Hence it might be interesting to even find an answer to the following question:

If contains a measure 1 set in every direction, does it follow that ?

Billiards

Amie Wilkinson asked me the following question some time ago:

Given a smooth convex Jordan curve , consider the billiard map , let be the projection.

a) If , does this imply is a circle?

(Yes, in fact we only need to fix the second component of points in for a chosen pair of points . Classical geometry)

b) If , does this imply is a circle?

(No, my example was a cute construction that attaches six circular arcs together)

c) What’s the smallest set s.t. if fixes the second component on then has to be a circle?

I am still thinking about c)…My guess is that any sub interval would work, and of course any dense subset inside a given set works equally well as the whole set…

But is it possible to have only finitely many angles? Maybe even two angles?