On variants of the Furstenberg set problem

Jonathan M. Fraser

doi:10.1090/proc/17521

On variants of the Furstenberg set problem

Jonathan M. Fraser^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

Given an integer d ≥ 2, s ∈ (0, 1], and t ∈ [0, 2(d − 1)], suppose a set X in ℝ^d has the following property: there is a collection of lines of packing dimension t such that every line from the collection intersects X in a set of packing dimension at least s. We show that such sets must have packing dimension at least max{s, t/2} and that this bound is sharp. In particular, the special case d = 2 solves a variant of the Furstenberg set problem for packing dimension. We also solve the upper and lower box dimension variants of the problem. In both of these cases the sharp threshold is max{s, t + 1 − d}.

Original language	English
Number of pages	11
Journal	Proceedings of the American Mathematical Society
Volume	Articles in press
Early online date	14 Jan 2026
DOIs	https://doi.org/10.1090/proc/17521
Publication status	E-pub ahead of print - 14 Jan 2026

Keywords

Furstenberg set
Box dimension
Packing dimension

Access to Document

10.1090/proc/17521

Fraser_2025_PAMS_Variants-Furstenberg-set-problem_AAM_CCBY
Copyright © 2025 the Authors. This work has been made available online in accordance with the University of St Andrews Open Access policy. This accepted manuscript is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. The final published version of this work is available at https://doi.org/10.1090/proc/17521
Accepted author manuscript, 321 KBLicence: CC BY

1 Finished
1 Active

A refined fourier analysis: A refined fourier analysis
Fraser, J. (PI)
The Leverhulme Trust
1/09/24 → 29/02/28
Project: Standard
Fourier analytic tachniques in finitie: Fourier analytic techniques in finite fields
Fraser, J. (PI)
EPSRC
1/05/24 → 30/04/25
Project: Standard

Cite this

@article{9ee1be47656f4ef6ad83bcd15674f27f,

title = "On variants of the Furstenberg set problem",

abstract = "Given an integer d ≥ 2, s ∈ (0, 1], and t ∈ [0, 2(d − 1)], suppose a set X in ℝd has the following property: there is a collection of lines of packing dimension t such that every line from the collection intersects X in a set of packing dimension at least s. We show that such sets must have packing dimension at least max\{s, t/2\} and that this bound is sharp. In particular, the special case d = 2 solves a variant of the Furstenberg set problem for packing dimension. We also solve the upper and lower box dimension variants of the problem. In both of these cases the sharp threshold is max\{s, t + 1 − d\}.",

keywords = "Furstenberg set, Box dimension, Packing dimension",

author = "Fraser, \{Jonathan M.\}",

note = "Funding: The author was financially supported by an EPSRC Standard Grant (EP/Y029550/1) and a Leverhulme Trust Research Project Grant (RPG-2023-281).",

year = "2026",

month = jan,

day = "14",

doi = "10.1090/proc/17521",

language = "English",

volume = "Articles in press",

journal = "Proceedings of the American Mathematical Society",

issn = "0002-9939",

publisher = "American Mathematical Society",

}

TY - JOUR

T1 - On variants of the Furstenberg set problem

AU - Fraser, Jonathan M.

N1 - Funding: The author was financially supported by an EPSRC Standard Grant (EP/Y029550/1) and a Leverhulme Trust Research Project Grant (RPG-2023-281).

PY - 2026/1/14

Y1 - 2026/1/14

N2 - Given an integer d ≥ 2, s ∈ (0, 1], and t ∈ [0, 2(d − 1)], suppose a set X in ℝd has the following property: there is a collection of lines of packing dimension t such that every line from the collection intersects X in a set of packing dimension at least s. We show that such sets must have packing dimension at least max{s, t/2} and that this bound is sharp. In particular, the special case d = 2 solves a variant of the Furstenberg set problem for packing dimension. We also solve the upper and lower box dimension variants of the problem. In both of these cases the sharp threshold is max{s, t + 1 − d}.

AB - Given an integer d ≥ 2, s ∈ (0, 1], and t ∈ [0, 2(d − 1)], suppose a set X in ℝd has the following property: there is a collection of lines of packing dimension t such that every line from the collection intersects X in a set of packing dimension at least s. We show that such sets must have packing dimension at least max{s, t/2} and that this bound is sharp. In particular, the special case d = 2 solves a variant of the Furstenberg set problem for packing dimension. We also solve the upper and lower box dimension variants of the problem. In both of these cases the sharp threshold is max{s, t + 1 − d}.

KW - Furstenberg set

KW - Box dimension

KW - Packing dimension

U2 - 10.1090/proc/17521

DO - 10.1090/proc/17521

M3 - Article

SN - 0002-9939

VL - Articles in press

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

ER -

On variants of the Furstenberg set problem

Abstract

Keywords

Access to Document

Fingerprint

Projects

A refined fourier analysis: A refined fourier analysis

Fourier analytic tachniques in finitie: Fourier analytic techniques in finite fields

Cite this