Counting spectral radii of matrices with positive entries

The sum-product conjecture of Erdős and Szemerédi states that, given a finite set A of positive numbers, one can find asymptotic lower bounds for max{|A+A|,|A⋅A|} of the order of |A|1+δ for every δ<1. In this paper we consider the set of all spectral radii of n×n matrices with entries in A, and find lower bounds for the cardinality of this set. In the case n=2, this cardinality is necessarily larger than max{|A+A|,|A⋅A|}.