Projections in vector spaces over finite fields

Abstract: 

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:

• if the tree has 'few' leaves then the number of leaves on the shadow is similar to the number of leaves on the tree at almost all times.
• if the tree has 'many' leaves then the shadow is a 'domain' at almost all times; that is, we can avoid strong sunshine or rain under this tree.