CSUN Algebra, Number Theory, and Discrete Mathematics Seminar

The existence of a set $A\subset \mathbb{Z}$ of positive upper Banach density such that $A-A:=\left\{m-n:m,n\in A,m>n\right\}$ does not contain a set of the form $E-E$ with $E$ piecewise syndetic is in essence the content of a popular result due to Kříž in 1987, in which he used a graph-theoretical approach. More recently, other proofs of this result have been given using combinatorial number theory. Our goal here is to show that a stronger result than the one given by Kříž can be obtained, and that this can be done via operator theory, namely using dynamics of linear operators on Banach spaces.