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

Building prescribed quantitative orbit equivalence with the group of integers