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 | } .