Building prescribed quantitative orbit equivalence with the group of integers

Two groups are orbit equivalent if they both admit an action on a same probability space that share the same orbits. In particular the Ornstein-Weiss theorem implies that all infinite amenable groups are orbit equivalent to the group of integers. To refine this notion between infinite amenable groups Delabie, Koivisto, Le Ma\^itre and Tessera introduced a quantitative version of orbit equivalence. They furthermore obtained obstructions to the existence of such equivalence using the isoperimetric profile. In this article we offer to answer the inverse problem (find a group being orbit equivalent to a prescribed group with prescribed quantification) in the case of the group of integers using the so called F{\o}lner tiling shifts introduced by Delabie et al. To do so we use the diagonal products defined by Brieussel and Zheng giving groups with prescribed isoperimetric profile.