Counting Spectral Radii of Matrices with Positive Entries

TL;DR

This paper investigates the diversity of spectral radii of matrices with positive entries from a set A, establishing lower bounds on their number and relating it to sum-product phenomena in additive combinatorics.

Contribution

It introduces bounds on the number of distinct spectral radii of matrices with entries in A, connecting spectral properties to sum-product conjectures.

Findings

Lower bounds for spectral radii cardinality for matrices with entries in A.

For 2x2 matrices, spectral radii count exceeds max of sum and product set sizes.

Links spectral radius diversity to sum-product conjecture in additive combinatorics.

Abstract

The sum-product conjecture of Erd\H os and Szemer\'edi states that, given a finite set $A$ of positive numbers, one can find asymptotic lower bounds for $\max\{|A+A|,|A\cdot A|\}$ of the order of $|A|^{1+\delta}$ for every $\delta <1$. In this paper we consider the set of all spectral radii of $n\times n$ 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 $\max\{|A+A|,|A\cdot A|\}$.