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Additive and Multiplicative Structure in Matrix Spaces

Published online by Cambridge University Press:  01 March 2007

MEI-CHU CHANG*
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521, USA (e-mail: [email protected])

Abstract

Let A be a set of N matrices. Let g(A) ≔ |A + A| + |A · A|, where A + A = {a1 + a2aiA} and A · A = {a1a2aiA} 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 : aV is commutative, allowing us to decompose as eigenspaces simultaneously, so we can finish the proof with induction and a variant of the Erdős–Szemerédi argument.

Type
Paper
Copyright
Copyright © Cambridge University Press 2006

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References

[1]Bilu, Y. (1999) Structure of sets with small sumset. Structure Theory of Set Addition: Astérisque 258 pp. 77108.Google Scholar
[2]Bourgain, J. and Chang, M.-C. (2004) On the size of k-fold sum and product sets of integers. J. Amer. Math. Soc. 17 pp. 473497.CrossRefGoogle Scholar
[3]Bourgain, J., Katz, N. and Tao, T. (2004) A sum-product estimate in finite fields, and applications. GAFA 14 pp. 2757.Google Scholar
[4]Chang, M.-C. (2002) Factorization in generalized arithmetic progressions and applications to the Erdős–Szemerédi sum-product problems. GAFA 113 pp. 399419.Google Scholar
[5]Chang, M.-C. (2003) Erdős–Szemerédi problem on sum set and product set. Annals of Math. 157 939957.CrossRefGoogle Scholar
[6]Chang, M.-C. (2004) On sums and products of distinct numbers. J. Combin. Theory Ser. A 105 349354.CrossRefGoogle Scholar
[7]Chang, M.-C. (2004) A sum-product theorem in semi-simple commutative Banach algebras. J. Funct. Anal. 212 399430.CrossRefGoogle Scholar
[8]Chang, M.-C. (2005) A sum-product estimate in algebraic division algebras over R. Israel J. Math. 150 pp. 369380.CrossRefGoogle Scholar
[9]Elekes, G. (1997) On the number of sums and products. Acta Arithmetica 81 365367.CrossRefGoogle Scholar
[10]Elekes, G. and Ruzsa, I. Z. (2003) Product sets are very large if sumsets are very small. Studia Sci. Math. Hungar. 40.Google Scholar
[11]Erdős, P. and Szemerédi, E. (1983) On sums and products of integers. In Studies in Pure Mathematics, Birkhäuser, Basel, pp. 213218.CrossRefGoogle Scholar
[12]Ford, K. (1998) Sums and products from a finite set of real numbers. Ramanujan J. 2 5966.CrossRefGoogle Scholar
[13]Nathanson,, M. B. (1993) The simplest inverse problems in additive number theory. In Number Theory with an Emphasis on the Markoff Spectrum (Pollington, A. and Moran, W., eds), Marcel Dekker, pp. 191206.Google Scholar
[14]Nathanson, M. B. (1996) Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Springer.CrossRefGoogle Scholar
[15]Nathanson, M. B. (1997) On sums and products of integers. Proc. Amer. Math. Soc. 125 916.CrossRefGoogle Scholar
[16]Nathanson, M. and Tenenbaum, G. (1999) Inverse theorems and the number of sums and products. Structure Theory of Set Addition: Astérisque 258 195204.Google Scholar
[17]Ruzsa, I. Z. (1994) Generalized arithmetic progressions and sumsets. Acta Math. Hungar. 65 379388.CrossRefGoogle Scholar
[18]Solymosi, J. (2005) On the number of sums and products. Bulletin London Math. Soc. (4) 37 pp. 491494.CrossRefGoogle Scholar