Formal squares over the unit square separate FS-domains from RB-domains

Abstract

Every RB-domain is an FS-domain. Whether the converse holds was a long-standing open problem. We prove that the domain of closed axis-parallel squares in the plane whose centres lie in the unit square $[0,1{]}^2$, with the whole plane adjoined and ordered by reverse inclusion, is an FS-domain but not an RB-domain. The proof is quantitative. A finite-grid argument first shows that, on every finite slab $[0,1{]}^2\times [0,m]$, for every approximate identity and every $\epsilon >0$, one member of the approximate identity has radius excess (i.e., the output radius minus the input radius) uniformly at most $\epsilon$. By contrast, for every deflation and every $m>0$, the radius excess is at least $m/(4m+1)$ at some point of the slab $[0,1{]}^2\times [0,m]$. This solution was obtained independently of the recent work of Chen, Kou, and Lyu and uses a different method.