Towards dimension expanders over finite fields

In this paper we study the problem of explicitly constructing a dimension expander raised by [3]: Let double-struck F signⁿ be the n dimensional linear space over the field double-struck F sign. Find a small (ideally constant) set of linear transformations from double-struck F sign ⁿ to itself {A_i}_iεI such that for every linear subspace V ⊂ double-struck F signⁿ of dimension dim(V) < n/2 we have dim (Σ_iεIA_i(V)) > (l+α) ·dim(V), where α > 0 is some constant. In other words, the dimension of the subspace spanned by {A_i(V)} _iεI should be at least (1 + α) · dim(V). For fields of characteristic zero Lubotzky and Zelmanov [8] completely solved the problem by exhibiting a set of matrices, of size independent of n, having the dimension expansion property. In this paper we consider the finite field version of the problem and obtain the following results. 1. We give a constant number of matrices that expand the dimension of every subspace of dimension d < n/2 by a factor of (1 + 1/log n). 2. We give a set of O (log n) matrices with expanding factor of (1 + α), for some constant α > 0. Our constructions are algebraic in nature and rely on expanding Cayley graphs for the group ℤ/ℤn and smalldiameter Cayley graphs for the group SL₂(p).