Building prescribed quantitative orbit equivalence with the integers (Q6594556)

\textit{D. S. Ornstein} and \textit{B. Weiss} [Bull. Am. Math. Soc., New Ser. 2, 161--164 (1980; Zbl 0427.28018)] showed that free actions of infinite countable amenable groups are all orbit equivalent to actions of the integers. Beyond amenable groups it is well-known that the situation is entirely different, and orbit equivalence becomes a highly non-trival relation. Here a different direction is pursued, with a contribution to refining orbit equivalence within the amenable setting building on work of \textit{T. Delabie} et al. [Ann. Henri Lebesgue 5, 1417--1487 (2022; Zbl 07646108)] that introduced a quantitative version of orbit equivalence. Here the inverse problem of finding a group action orbit equivalent to a given group with prescribed quantification of the orbit equivalence is attacked using `Følner tiling sets' and using the diagonal products defined by \textit{J. Brieussel} and \textit{T. Zheng} [Ann. Math. (2) 193, No. 1, 1--105 (2021; Zbl 1535.20232)] to find groups with prescribed isoperimetric profile.