Recently there has been a growing interest in studying finite field versions of some classical problems arising from Euclidean spaces. In this talk we study the finite field version of a basic problem in fractal geometry: how the projections affect dimension. Among other things, we obtain the Marstand-Mattila type projection theorem in finite fields.
Roughly speaking: given a tree, we observe the shadow of this tree from morning to night. The question is what is the 'size' of the shadow. The Marstand-Mattila projection theorem provides an answer: