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Additive and Multiplicative Structure in Matrix Spaces.
- Source :
- Combinatorics, Probability & Computing; Mar2007, Vol. 16 Issue 2, p219-238, 20p
- Publication Year :
- 2007
-
Abstract
- Let $A$ be a set of $N$ matrices. Let $g(A)\ccolon=|A+A|+|A\cdot A|$, where $A+A=\{a_1 + a_2 \mid a_i \in A\}$ and $A\cdot A=\{a_1a_2 \mid a_i \in A\}$ 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 $d\times d$ symmetric matrices, then there exists $\varepsilon=\varepsilon (d)>0$ such that $g(A)>N^{1+\varepsilon}.$ 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 $\{a^2\ccolon a\in V\}$ 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 09635483
- Volume :
- 16
- Issue :
- 2
- Database :
- Complementary Index
- Journal :
- Combinatorics, Probability & Computing
- Publication Type :
- Academic Journal
- Accession number :
- 24080009
- Full Text :
- https://doi.org/10.1017/S0963548306008145