A question by Furstenberg

September 25, 2009

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 $S \in \mathbb{R}^2$, if $\exists \delta>0$ s.t. for all direction $\theta, \exists$ line $l$ with direction $\theta$ s.t. $\dim (l \cap S) > \delta$. Does it follow that $\dim(A) \geq 1$?

Note that a stronger conjecture would be $\dim(A)$ is at least $1+\delta$ which when taking $\delta = 1$ would give a generalization of the $2$-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 $A \subseteq \mathbb{R}^2$ contains a measure 1 set in every direction, does it follow that $\dim(A)=2$?