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PreprintarXiv licencearXiv

Spread-out Furstenberg sets II: general configurations and fractal restricted projections

  • Alex Cohen
  • Pablo Shmerkin
  • Hong Wang
  • Dmitrii Zakharov
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Abstract

We prove optimal dual Furstenberg estimates for Frostman and Katz--Tao configurations for which the shadings are not locally concentrated near pencils. The starting point is the sticky estimate of Orponen and Shmerkin in a companion paper. We deduce restricted projection theorems in the plane and in higher dimensions. In particular, Ahlfors-regular measures on the circle satisfy a Marstrand-type projection theorem. Applications include restricted projections to surfaces in the 𝔾(4,2) Grassmannian, pinned distances, discretized sum-product, and a new proof of the dimension formula for planar self-affine sets.

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Cite

BibTeXRIS
BibTeX
@misc{cohen2026spreadout,
  title = {{Spread-out Furstenberg sets II: general configurations and fractal restricted projections}},
  author = {Alex Cohen and Pablo Shmerkin and Hong Wang and Dmitrii Zakharov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13682},
  eprint = {2610.13682},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13682},
}
RIS
TY  - UNPB
TI  - Spread-out Furstenberg sets II: general configurations and fractal restricted projections
AU  - Alex Cohen
AU  - Pablo Shmerkin
AU  - Hong Wang
AU  - Dmitrii Zakharov
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13682
UR  - https://arxiv.org/abs/2610.13682
AB  - We prove optimal dual Furstenberg estimates for Frostman and Katz--Tao configurations for which the shadings are not locally concentrated near pencils. The starting point is the sticky estimate of Orponen and Shmerkin in a companion paper. We deduce restricted projection theorems in the plane and in higher dimensions. In particular, Ahlfors-regular measures on the circle satisfy a Marstrand-type projection theorem. Applications include restricted projections to surfaces in the 𝔾(4,2) Grassmannian, pinned distances, discretized sum-product, and a new proof of the dimension formula for planar self-affine sets.
ER  -