Asymptotic average shadowing property on compact metric spaces

We prove that if for a continuous map f$f$ on a compact metric space X$X$, the chain recurrent set, R(f)$R\left(f\right)$ has more than one chain component, then f$f$ does not satisfy the asymptotic average shadowing property. We also show that if a continuous map f$f$ on a compact metric space X$X$ has the asymptotic average shadowing property and if A$A$ is an attractor for f$f$, then A$A$ is the single attractor for f$f$ and we have A=R(f)$A=R\left(f\right)$. We also study diffeomorphisms with asymptotic average shadowing property and prove that if M$M$ is a compact manifold which is not finite with dimM=2$dimM=2$, then the C¹$C^1$ interior of the set of all C¹$C^1$ diffeomorphisms with the asymptotic average shadowing property is characterized by the set of Ω$\Omega$-stable diffeomorphisms.     

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|}$max\{|A+A|,|A\cdot A|\}$.     

On the exterior algebra method applied to restricted set addition

In 1994 Dias da Silva and Hamidoune solved a long-standing open problem of Erd?s and Heilbronn using the structure of cyclic spaces for derivatives on Grassmannians and the representation theory of symmetric groups. They proved that for any subset A$A$ of the p$p$-element group Z/pZ$\mathbb{Z}/p\mathbb{Z}$ (where p$p$ is a prime), at least min{p,m|A|−m²+1}$min\{p,m\left|A\right|-m^2+1\}$ different elements of the group can be written as the sum of m$m$ different elements of A$A$. In this note we present an easily accessible simplified version of their proof for the case m=2$m=2$, and explain how the method can be applied to obtain the corresponding inverse theorem.