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, $α$ be a positive number and $( β_j)_{j \geqslant 1}$ be a finitely supported sequence of non-negative numbers. We show that there is a quantitatively $t^α\cdot \prod_{j \geqslant 1} ( \log^{(j)}{t} )^{-β_j} $-integrable measure equivalence from $\mathbb{Z}^n$ to $\mathbb{Z}^m$ if and only if either $α<m/n$ or $α=m/n$ and there is $j_* \geqslant 1$ such that $β_j= 1$ for all $1 \leqslant j < j_*$ and $β_{j_*} > 1$. 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 $t^α$-integrable ($α>0$) measure equivalence (or orbit equivalence) from $\mathbb{Z}^n$ to $\mathbb{Z}^m$ if and only if $α<m/n$. 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.