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Bulletin of the London Mathematical Society Advance Access originally published online on July 19, 2009
Bulletin of the London Mathematical Society 2009 41(5):817-822; doi:10.1112/blms/bdp054
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© 2009 London Mathematical Society

Sum–product estimates for well-conditioned matrices

J. Solymosi

Department of Mathematics
University of British Columbia
1984 Mathematics Road
Vancouver, BC
CanadaV6T 1Z2

V. Vu

Department of Mathematics
Rutgers University
110 Frelinghuysen Road
Piscataway, NJ 08554
USA
vanvu@math.rutgers.edu

Received 12 February 2008. Revision received 9 April 2009.

We show that if A is a finite set of d x d well-conditioned matrices with complex entries, then the following sum–product estimate holds | A + A | x |A·A| = {Omega} (|A| 5/2).

Dedicated to the memory of György Elekes

2000 Mathematics Subject Classification 11B75 (primary), 15A45, 11C20 (secondary).

The research was conducted while both researchers were members of the Institute for Advanced Study. Funding provided by The Charles Simonyi Endowment. The first author was supported by NSERC and OTKA grants and by Sloan Research Fellowship. The second author was supported by an NSF Career Grant.


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