Towards dimension expanders over finite fields

Abstract

In this paper we study the problem of explicitly constructing a dimension expander raised by [3]: Let $\mathbb{F}^n$ be the n dimensional linear space over the field $\mathbb{F}$. Find a small (ideally constant) set of linear transformations from $\mathbb{F}^n$ to itself {A _i } _i∈I such that for every linear subspace V ⊂ $\mathbb{F}^n$ of dimension dim(V)<n/2 we have

$\dim \left( {\sum\limits_{i \in I} {A_i (V)} } \right) \geqslant (1 + \alpha ) \cdot \dim (V),$

where α>0 is some constant. In other words, the dimension of the subspace spanned by {A _i (V)} _i∈I should be at least (1+α)·dim(V). For fields of characteristic zero Lubotzky and Zelmanov [10] 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/logn).

• 2. We give a set of O<(logn) 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 small-diameter Cayley graphs for the group SL₂(p).