Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2507.09605

A bound for plany Kakeya sets in $\mathbb{F}_q^4$ using the planebrush method

Izabella Łaba·Mukul Rai Choudhuri·Joshua Zahl
View original source ↗Read source document ↗Export metadata ↓

Abstract

Katz and Zahl used a planebrush argument to prove that Kakeya sets in $\mathbb{R}^4$ have Hausdorff dimension at least 3.059. In the special case when the Kakeya set is plany, their argument gives a better lower bound of 10/3. We give a nontechnical exposition of the Katz-Zahl argument for plany Kakeya sets in the finite field setting.

Explore related subjects

math.CAmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Izabella Łaba, Mukul Rai Choudhuri, Joshua Zahl. 2025-07-13. A bound for plany Kakeya sets in $\mathbb{F}_q^4$ using the planebrush method. https://doi.org/10.1007/s40687-025-00594-8

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the prime field spherical restriction conjecture in four dimensions: breaking the Stein-Tomas exponent and applications

Let $p$ be an odd prime. We prove the extension estimate $R_{S_j}^*(2\to r)\lesssim_r 1$ for every nonzero-radius sphere $S_j\subseteq\mathbb{F}_p^4$ and every $r\geq \, 34/11$, uniformly in $p$ and $j$. This improves the Stein--Tomas exponent $10/3$ established by Iosevich and Koh (2008). We also formulate a localized spherical restriction/extension conjecture that predicts the sharp dependence of the restriction norm on the size of the physical support. This conjecture implies the spherical extension estimates $R_{S_j}^*(2\to r)\lesssim_r 1$ for every $r>3$, and yields almost-every-pin distance estimates at the conjectured Erdős--Falconer exponent in four dimensions, up to an arbitrarily small power loss in the set-size hypothesis. Using the same method, we improve the bounds supplied by Fourier decay and Plancherel at intermediate support scales and derive new almost-every-pin distance estimates in $\mathbb{F}_p^4$.

math.CA↗
arXiv2026

Dimension-free estimates for discrete maximal functions over cubes in $\mathbb Z^d$

In this short note, we establish dimension-free $\ell^p(\mathbb Z^d)$ bounds, for all $p\in(1,\infty]$, for the discrete Hardy--Littlewood maximal functions associated with cubes in $\mathbb Z^d$, answering a question that had been open for a while. The key idea is to prove dimension-free bounds for the $\ell^p(\mathbb Z^d)$ norms of the differences of the corresponding averages. This follows from an ad hoc interpretation of the associated discrete multipliers as a special continuous family of multipliers to which basic fractional integration and complex interpolation can be applied. The same method also yields an elementary proof of Bourgain's dimension-free $L^p(\mathbb R^d)$ bounds for the Hardy--Littlewood maximal function associated with cubes in $\mathbb R^d$.

math.CA↗
arXiv2026

Establishing the Polynomial Wolff Axioms for $δ$-Separated $δ$-Tubes With #o-minimality

We establish the full version of a conjecture of Guth and Zahl, giving a lower bound for the volume of a semialgebraic set that has a large intersection with a collection of $δ$-separated $δ$-tubes. Our proof uses o-minimal methods to simplify the proof of Katz and Rogers, who proved the conjecture up to a small factor. We also establish that the constants depend polynomially on the complexity of the semialgebraic set, and more generally in the #o-minimal setting.

math.CA↗