Dimension expanders

TL;DR

This paper constructs explicit linear transformations over fields of characteristic zero that significantly expand subspaces, answering a question by Avi Wigderson and leaving open the positive characteristic case.

Contribution

It introduces explicit linear transformations that achieve subspace expansion over characteristic zero fields, advancing understanding of dimension expanders.

Findings

Existence of explicit dimension expanders over characteristic zero fields.

Quantitative bounds on subspace expansion for small subspaces.

Addresses a previously open question by Avi Wigderson.

Abstract

We show that there exists $k \in \bbn$ and $0 < \e \in\bbr$ such that for every field $F$ of characteristic zero and for every $n \in \bbn$, there exists explicitly given linear transformations $T_1,..., T_k: F^n \to F^n$ satisfying the following: For every subspace $W$ of $F^n$ of dimension less or equal $\frac n2$, $ \dim(W+\suml^k_{i=1} T_iW) \ge (1+\e) \dim W$. This answers a question of Avi Wigderson [W]. The case of fields of positive characteristic (and in particular finite fields) is left open.