Spread-out Furstenberg sets I: the sticky case
Abstract
We introduce spread-out measures, a notion of local non-concentration near pencils for measures on Grassmannians. We establish an optimal δ-discretized dual Furstenberg estimate for ``sticky spread-out'' shadings. A simple-to-state corollary is the following. Assume that ν is an Ahlfors s-regular measure on 𝔾(2,1) with s > 0. Let E ⊂ ℝ2 be a Borel set with equal Hausdorff and packing dimension. Then, $\dim_{\mathrm{H}} Π_{V}(E) = \min\{\dim_{\mathrm{H}} E,1\} \quad \text{for ν-a.e. } V \in S^1$. The general δ-discretized estimate for ``sticky spread out'' configurations here is the starting point for a more general theorem in the companion paper of Cohen, Shmerkin, Wang and Zakharov, where the ``stickiness'' hypothesis is removed. For instance, the assumption of equal Hausdorff and packing dimension in the result stated above will be dispensed with in the companion paper.
Related
- Kakeya and Furstenberg problems for sticky sets of tubes
- Spread-out Furstenberg sets II: general configurations and fractal restricted projections
- The local dynamical structure of Δ* sets via a new Furstenberg family algebra
- Hyperplane Furstenberg Sets in Higher Dimensions
- Sticky Brownian Motion with a Locally Finite Atomic Sticky Measure
BibTeX
@misc{orponen2026spreadout,
title = {{Spread-out Furstenberg sets I: the sticky case}},
author = {Tuomas Orponen and Pablo Shmerkin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13677},
eprint = {2610.13677},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13677},
}
RIS
TY - UNPB
TI - Spread-out Furstenberg sets I: the sticky case
AU - Tuomas Orponen
AU - Pablo Shmerkin
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13677
UR - https://arxiv.org/abs/2610.13677
AB - We introduce spread-out measures, a notion of local non-concentration near pencils for measures on Grassmannians. We establish an optimal δ-discretized dual Furstenberg estimate for ``sticky spread-out'' shadings. A simple-to-state corollary is the following. Assume that ν is an Ahlfors s-regular measure on 𝔾(2,1) with s > 0. Let E ⊂ ℝ^2 be a Borel set with equal Hausdorff and packing dimension. Then, $\dim_{\mathrm{H}} Π_{V}(E) = \min\{\dim_{\mathrm{H}} E,1\} \quad \text{for ν-a.e. } V \in S^1$. The general δ-discretized estimate for ``sticky spread out'' configurations here is the starting point for a more general theorem in the companion paper of Cohen, Shmerkin, Wang and Zakharov, where the ``stickiness'' hypothesis is removed. For instance, the assumption of equal Hausdorff and packing dimension in the result stated above will be dispensed with in the companion paper.
ER -