Counting spectral radii of matrices with positive entries

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