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
?