Forbidden Intersection Problems for Families of Linear Maps

We study an analogue of the Erdős-Sós forbidden intersection problem, for families of invertible linear maps. If V and W are vector spaces over the same field, we say a family F of linear maps from V to W is (t −1)-intersection-free if for any two linear maps σ1,σ2 ∈ F, the dimension of the subspace {v ∈ V : σ1(v) = σ2(v)} is not equal to t −1. We prove that if n is sufficiently large depending on t, q is any prime power, V is an n-dimensional vector space over Fq, and F ⊂ GL(V) is (t −1)-intersection-free, then Equality holds only if there exists a t-dimensional subspace of V on which all elements of F agree, or a t-dimensional subspace of V∗ on which all elements of {σ∗ : σ ∈ F} agree. Our main tool is a ‘junta approximation’ result for families of linear maps with a forbidden intersection: namely, that if V and W are finite-dimensional vector spaces over the same finite field, then any (t−1)-intersection-free family of linear maps from V toW is essentially contained in a t-intersecting junta (meaning, a family J of linear maps from V toW such that the membership of σ in J is determined by σ(v1), …,σ(vM),σ∗(a1), …,σ∗(aN), where v1, …,vM ∈ V, a1, …,aN ∈W∗ and M+N is bounded). The proof of this in turn relies on a variant of the ‘junta method’ (originally introduced by Dinur and Friedgut [6] and powerfully extended by Keller and the last author [16]), together with spectral techniques and a new hypercontractive inequality.