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Additive and Multiplicative Structure in Matrix Spaces

Published online by Cambridge University Press:  01 March 2007

MEI-CHU CHANG
MEI-CHU CHANG*
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521, USA (e-mail: mcc@math.ucr.edu)
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Abstract

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Let A be a set of N matrices. Let g(A) ≔ |A + A| + |A · A|, where A + A = {a1 + a2 ∣ ai ∈ A} and A · A = {a1a2 ∣ ai ∈ 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)>N1+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 a2 : 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.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 16 , Issue 2 , March 2007 , pp. 219 - 238
Copyright
Copyright © Cambridge University Press 2006

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