Additive and Multiplicative Structure in Matrix Spaces

Source abstract

Let A be a set of N matrices. Let g ( A ) ≔ | A + A | + | A · A |, where A + A = { a 1 + a 2 ∣ a i ∈ A } and A · A = { a 1 a 2 ∣ a i ∈ A } are the sum set and product set. We prove that if the determinant of the difference of any two distinct matrices in A is nonzero, then g ( A ) cannot be bounded below by cN for any constant c . We also prove that if A is a set of d × d symmetric matrices, then there exists ϵ = ϵ( d )>0 such that g ( A )> N 1+ϵ . For the first result, we use the bound on the number of factorizations in a generalized progression. For the symmetric case, we use a technical proposition which provides an affine space V containing a large subset E of A , with the property that if an algebraic property holds for a large subset of E , then it holds for V . Then we show that the system a 2 : a ∈ V is commutative, allowing us to decompose ${\mathbb R}^d$ as eigenspaces simultaneously, so we can finish the proof with induction and a variant of the Erdős–Szemerédi argument.