Bull. London Math. Soc. 41 (2009) 817–822 C ❡ 2009 London Mathematical Society doi:10.1112/blms/bdp054 Sum–product estimates for well-conditioned matrices J. Solymosi and V. Vu Dedicated to the memory of Gy¨orgy Elekes Abstract We show that if A is a finite set of d × d well-conditioned matrices with complex entries, then the following sum–product estimate holds |A + A| × |A · A| = Ω(|A| 5/2 ). 1. Introduction Let A be a finite subset of a ring Z . The sum–product phenomenon, first investigated by Erd˝ os and Szemer´ edi [4], suggests that either A·A or A + A is much larger than A. This was first proved for Z, the ring of integers, in [4]. Recently, many researchers have studied (with considerable success) other rings. Several of these results have important applications in various fields of mathematics. The interested readers are referred to Bourgain’s survey [1]. In this paper we consider Z being the ring of d × d matrices with complex entries. (We are going to use the notation ‘matrix of size d’ for d × d matrices.) It is well known that one cannot generalize the sum–product phenomenon, at least in the straightforward manner, in this case. The archetypal counterexample is the following: Example 1.1. Let I denote the identity matrix and let E ij be the matrix with only one non- zero entry at position ij and this entry is one. Let M a := I + aE 1d and let A = {M 1 ,...,M n }. It is easy to check that |A + A| = |A · A| =2n − 1. This example suggests that one needs to make some additional assumptions in order to obtain a non-trivial sum–product estimate. Chang [2] proved the following Theorem 1.2. There is a function f = f (n) tending to infinity with n such that the following holds. Let A be a finite set of matrices of size d over the reals such that for any M = M ′ ∈A, we have det(M − M ′ ) =0. Then we have |A + A| + |A · A|  f (|A|)|A|. The function f in Chang’s proof tends to infinity slowly. In most applications, it is desirable to have a bound of the form |A| 1+c for some positive constant c. In this paper, we show that this is indeed the case (and in fact c can be set to be 1 4 ) if we assume that the matrices are far from being singular. Furthermore, this result provides a new insight into the above counterexample (see the discussion following Theorem 2.2). Received 12 February 2008; revised 9 April 2009; published online 19 July 2009. 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.