Kriz’s theorem via dynamics of linear operators (Yunied Puig de Dios, CMC)

The existence of a set $A\subset \N_0$ of positive upper Banach density such that $A-A:=\{m-n:m, n\in A, m>n\}$ does not contain a set of the form $S-S$ with $S$ a piecewise syndetic is in essence the content of a popular result due to K\v r\'{i}\v z in 1987. Since then at least four different proofs of this result have been given, and all of them give basically the example originally exhibited by K\v r\'{i}\v z when viewed appropriately. We obtain a generalization of K\v r\'{i}\v z’s result. Our approach differs completely from the previous ones, as this would be the first proof of K\v r\'{i}\v z’s Theorem which does not rely on Lov\'{a}sz’s Theorem for chromatic numbers of Kneser graphs. Furthermore, it is done via operator theory, namely using dynamics of bounded linear operators on infinite-dimensional complex separable Banach spaces. As a consequence, our example is genuinely different from the one exhibited  originally by K\v r\'{i}\v z.