Distinguishing sets and measures from their planar Brownian images
Abstract
Motivated by a question of Hochman, we investigate whether two deterministic time measures or time sets can be distinguished almost surely from a single planar Brownian image. Using rescaled correlation integrals, we give a criterion for mutual singularity of the laws of the image measures, even when the time measures have the same dimension. For self-similar time sets satisfying the open set condition, we also recover the natural occupation measure from the compact image. Together, these results allow us to distinguish lattice and non-lattice self-similar sets and their natural measures within this class. For a naturally parametrised family with two contractions, with fixed dimension, each parameter can be distinguished from all but finitely many others. If the dimension is sufficiently small, every distinct parameter can be distinguished. The results are formalised in Lean.
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BibTeX
@misc{troscheit2026distinguishing,
title = {{Distinguishing sets and measures from their planar Brownian images}},
author = {Sascha Troscheit},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16339},
eprint = {2610.16339},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16339},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Distinguishing sets and measures from their planar Brownian images
AU - Sascha Troscheit
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.16339
UR - https://arxiv.org/abs/2610.16339
AB - Motivated by a question of Hochman, we investigate whether two deterministic time measures or time sets can be distinguished almost surely from a single planar Brownian image. Using rescaled correlation integrals, we give a criterion for mutual singularity of the laws of the image measures, even when the time measures have the same dimension. For self-similar time sets satisfying the open set condition, we also recover the natural occupation measure from the compact image. Together, these results allow us to distinguish lattice and non-lattice self-similar sets and their natural measures within this class. For a naturally parametrised family with two contractions, with fixed dimension, each parameter can be distinguished from all but finitely many others. If the dimension is sufficiently small, every distinct parameter can be distinguished. The results are formalised in Lean.
ER -