  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.
