Sharp iterated-Logarithmic thresholds for quantitative measure and orbit equivalence between integer lattices

We identify the sharp iterated-logarithmic thresholds at the critical exponent for quantitative measure equivalence and orbit equivalence between integer lattices. To be more precise, let $n>m$ be two positive integers, $\alpha$ be a positive number and $({\beta }_j{)}_{j⩾1}$ be a finitely supported sequence of non-negative numbers. We show that there is a quantitatively $t^{\alpha }\cdot {\prod }_{j⩾1}({\log }^{(j)}t{)}^{-{\beta }_j}$-integrable measure equivalence from ${\mathbb{Z}}^n$ to ${\mathbb{Z}}^m$ if and only if either $\alpha <m\mathrm{/}n$ or $\alpha =m\mathrm{/}n$ and there is $j_{\ast }⩾1$ such that ${\beta }_j=1$ for all $1⩽j<j_{\ast }$ and ${\beta }_{j_{\ast }}>1$. Here ${\log }^{(j)}$ is the $j$-fold iterated logarithm. The same characterization holds for quantitative orbit equivalence. This characterization greatly strengthens the previous best-known result, due to the work of Delabie, Koivisto, Le Maître and Tessera (2022) and the work of Correia (2025), which asserts that there is a $t^{\alpha }$-integrable ($\alpha >0$) measure equivalence (or orbit equivalence) from ${\mathbb{Z}}^n$ to ${\mathbb{Z}}^m$ if and only if $\alpha <m\mathrm{/}n$. In particular, our result solves, in much stronger forms, two open problems posed respectively by Delabie, Koivisto, Le Maître and Tessera, and by Naryshkin and Petrakos.