整数格之间定量测度等价与轨道等价的尖锐迭代对数阈值
Sharp iterated-Logarithmic thresholds for quantitative measure and orbit equivalence between integer lattices
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中文总结 AI 辅助
本文确定了整数格之间定量测度等价与轨道等价的尖锐迭代对数阈值，给出了精确的充要条件，并大大加强了先前结果，解决了两个开放问题。
中文摘要 AI 辅助
我们确定了整数格之间定量测度等价与轨道等价在临界指数处的尖锐迭代对数阈值。更精确地说，设$n>m$为两个正整数，$\alpha$为正数，$( \beta_j)_{j \geqslant 1}$为有限支撑的非负数列。我们证明，存在从$\mathbb{Z}^n$到$\mathbb{Z}^m$的定量$t^{\alpha}\cdot \prod_{j \geqslant 1} ( \log^{(j)}{t} )^{-\beta_j} $-可积测度等价，当且仅当要么$\alpha<m/n$，要么$\alpha=m/n$且存在$j_* \geqslant 1$使得对所有$1 \leqslant j < j_*$有$\beta_j= 1$且$\beta_{j_*} > 1$。这里$\log^{(j)}$表示$j$重迭代对数。同样的刻画对定量轨道等价也成立。这一刻画大大加强了先前已知的最佳结果，该结果归功于Delabie、Koivisto、Le Maître和Tessera（2022年）的工作以及Correia（2025年）的工作，他们断言存在从$\mathbb{Z}^n$到$\mathbb{Z}^m$的$t^{\alpha}$-可积（$\alpha>0$）测度等价（或轨道等价）当且仅当$\alpha<m/n$。特别地，我们的结果以更强的形式解决了分别由Delabie、Koivisto、Le Maître和Tessera以及Naryshkin和Petrakos提出的两个开放问题。
英文摘要
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.
