Sharp iterated-Logarithmic thresholds for quantitative measure and orbit equivalence between integer lattices
Abstract
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 ( βj)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 ℤn to ℤm if and only if either α<m/n or α=m/n and there is j* geqslant 1 such that βj= 1 for all 1 leqslant j < j* and βj* > 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 tα-integrable (α>0) measure equivalence (or orbit equivalence) from ℤn to ℤm 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.
Related
- Iterated Graph Systems (III): Alexander-Orbach conjecture on hierarchical lattices
- Iterated Graph Systems (II): Bernoulli percolation and critical-exponent universality classes on hierarchical lattices
- The equivalence of solutions to fractional and logarithmic Helmholtz equations
- Quantitative orbit equivalence for rank-one systems
- Quantitative analysis of ground states for the fractional logarithmic Schrödinger equation
BibTeX
@misc{jiao2026sharp,
title = {{Sharp iterated-Logarithmic thresholds for quantitative measure and orbit equivalence between integer lattices}},
author = {Changhua Jiao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15816},
eprint = {2609.15816},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15816},
}
RIS
TY - UNPB
TI - Sharp iterated-Logarithmic thresholds for quantitative measure and orbit equivalence between integer lattices
AU - Changhua Jiao
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15816
UR - https://arxiv.org/abs/2609.15816
AB - 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 ℤ^n to 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 ℤ^n to 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.
ER -