The Kakeya problem in finite fields

The kakeya problem, in its best known formulation, is the following. Let E\subset\mathbf{R}^n be set which contains a translate of every unit segment; equivalently, for every direction e\in S^{n-1}, E contains a unit line segment parallel to e. An n-dimensional ball of radius 1/2 is a simple example of  a set with this property, but there are many other such sets, some of which have n-dimensional measure 0.

Can E be even smaller than that and have Hausdorff dimension strictly smaller that n?

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