Beating the elementary bound $n-\dim_{\mathrm{P}}A$ for $A$-restricted Kakeya sets when the midpoint set has small packing dimension

A Kakeya set in $\mathbb{R}^n$ is a compact set containing a unit line segment in every direction, and the Kakeya conjecture asserts that such sets have Hausdorff dimension $n$; this is open for $n\ge 4$. Fraser and Yang (arXiv:2505.05709, 2025) introduced $A$-restricted Kakeya sets, where the midpoint of the segment in each direction must lie in a fixed set $A$ with packing dimension at most $s$, and proved the elementary bound $\dim_{\mathrm{H}}K_A\ge n-s$ together with bush-argument improvements $\max\{n-s,\,n-g_n(s)\}$ (for example $\max\{4-s,\,19/5-3s/5\}$ in $\mathbb{R}^4$). They explicitly state that they do not know how to beat the bound $n-\dim_{\mathrm{P}}A$ when $\dim_{\mathrm{P}}A$ is close to zero. The open question for dimensions $n\ge 4$ is to determine the local behavior of the optimal lower bound $F_n(s)$ near $s=0$ — either prove an explicit improvement $\dim_{\mathrm{H}}K_A\ge n-s+\omega(s)$ with $\omega(s)>0$ for all sufficiently small $s>0$, or construct $A$-restricted Kakeya sets showing that $n-s$ is essentially sharp as $s\to 0$. No later work addresses this small-$s$ dimension regime: a subsequent paper on restricted Kakeya sets (Chen, Li, Li, Liu, Sun, Wang and Zhou, arXiv:2609.06556, September 2026) builds measure-zero examples with midpoints in every uncountable Borel set but contains no Hausdorff-dimension bounds, so the question remains open.

Background

A Kakeya set (or Besicovitch set) in $\mathbb{R}^n$ is a compact set containing a unit line segment in every direction $e\in S^{n-1}$. Besicovitch showed such sets can have Lebesgue measure zero, so one probes their fineness through the Hausdorff dimension $\dim_{\mathrm{H}}$ (defined via covers by sets of small diameter) and the packing dimension $\dim_{\mathrm{P}}$ (equivalently, the infimum over countable covers of the supremum of upper box-counting dimensions of the pieces, where upper box-counting dimension measures the growth rate of the minimal number of $\delta$-balls needed to cover the set). The Kakeya conjecture states that every Kakeya set in $\mathbb{R}^n$ has Hausdorff dimension $n$. It is a theorem in the plane (Davies, 1971) and, after the 2025 breakthrough of Hong Wang and Joshua Zahl (arXiv:2502.17655), in $\mathbb{R}^3$; it remains open in all dimensions $n\ge 4$, where the best known lower bound in $\mathbb{R}^4$ is $3.059$ (Katz and Zahl's planebrush estimate). A central technique is the Kakeya maximal function $f^*_\delta(e)=\sup_a \lvert T^\delta_e(a)\rvert^{-1}\int_{T^\delta_e(a)}\lvert f\rvert$, the supremum of averages of $f$ over $\delta$-tubes $T^\delta_e(a)$ (the $\delta$-neighborhoods of unit segments); Bourgain's bush argument exploits the fact that many tubes passing through a common point force a large union.

Fraser and Yang (arXiv:2505.05709) introduced a restricted variant: given a bounded set $A\subseteq\mathbb{R}^n$, a compact set $K_A$ is an $A$-restricted Kakeya set if for every direction $e\in S^{n-1}$ it contains a unit segment $I_e(a_e)$ in direction $e$ whose midpoint $a_e$ lies in $A$. Since $K_A-A$ contains the ball $B(0,1/2)$ and is a Lipschitz image of $K_A\times A$, the standard product inequality $\dim_{\mathrm{H}}(K_A\times A)\le \dim_{\mathrm{H}}K_A+\dim_{\mathrm{P}}A$ gives the elementary bound $\dim_{\mathrm{H}}K_A\ge n-\dim_{\mathrm{P}}A$. The same paper proves, via an $A$-restricted Kakeya maximal function and Bourgain's bush argument, the improved bound $\dim_{\mathrm{H}}K_A\ge\max\{n-s,\,n-g_n(s)\}$ whenever $\dim_{\mathrm{P}}A\le s$, where $g_n(s)$ is determined by any available Kakeya maximal estimate in $\mathbb{R}^{n-1}$ (Theorem 2.6); for instance, in $\mathbb{R}^4$ using Wolff's $\mathbb{R}^3$ maximal estimate one gets $\dim_{\mathrm{H}}K_A\ge\max\{4-s,\,19/5-3s/5\}$, and for $n\ge 5$ the Hickman–Rogers–Zhang higher-dimensional maximal estimates yield explicit piecewise bounds. The paper also shows the results survive replacing midpoints by an arbitrary point of each segment and upper box dimension by packing dimension.

Two structural facts frame the open question. First, since an $A$-restricted Kakeya set is in particular a Kakeya set, the unrestricted bounds apply: in $\mathbb{R}^2$ and $\mathbb{R}^3$ the dimension is always $n$, so the restricted problem lives entirely in $n\ge 4$. Second, at the endpoint $s=0$ the elementary bound is exact: if $\dim_{\mathrm{P}}A=0$ then $\dim_{\mathrm{H}}K_A\ge n$, hence $=n$. Nevertheless, in $\mathbb{R}^4$ the elementary term $4-s$ is the strongest of all known lower bounds precisely on the interval $0\le s<1/2$ (it also dominates the unrestricted 3.059 for $s<0.941$), because the bush-lifted term degenerates as $s\to 0$. This is the regime the authors flag: they write that they do not know how to beat the bound $n-\dim_{\mathrm{P}}A$ for $\dim_{\mathrm{P}}A$ close to zero. Whether the midpoint restriction to a small packing-dimension set forces dimension strictly above $n-s$ — or whether $n-s$ is the correct order — is unknown, and both known mechanisms (the covering/product argument and the bush lifting) are provably unable to settle it.

Problem Statement

Fix an integer $n\ge 4$. For a bounded set $A\subseteq\mathbb{R}^n$ with packing dimension at most $s\in[0,n]$, let $K_A$ be an $A$-restricted Kakeya set: a compact set containing, for every direction $e\in S^{n-1}$, a unit line segment in direction $e$ whose midpoint lies in $A$. Define

$F_n(s)=\inf\bigl\{\dim_{\mathrm{H}}K_A:\ \dim_{\mathrm{P}}A\le s,\ K_A\ \text{an}\ A\text{-restricted Kakeya set in }\mathbb{R}^n\bigr\}.$

The elementary bound gives $F_n(s)\ge n-s$, with equality at the endpoint $s=0$ (where $F_n(0)=n$), and the bush-lifted bounds of Fraser and Yang give no improvement over $n-s$ when $s$ is small. The problem posed by Fraser and Yang (arXiv:2505.05709), which they state they cannot resolve, is:

Is the elementary bound $n-\dim_{\mathrm{P}}A$ beatable when $\dim_{\mathrm{P}}A$ is close to zero?

For any single dimension $n\ge 4$, the question is resolved in one of two ways:

1. Improvement. Exhibit $s_0>0$ and an explicit function $\omega:(0,s_0)\to(0,\infty)$, and prove that every $A$-restricted Kakeya set $K_A$ in $\mathbb{R}^n$ with $\dim_{\mathrm{P}}A\le s$ satisfies $\dim_{\mathrm{H}}K_A\ge n-s+\omega(s)$ for all $0<s<s_0$.

2. Sharpness. Prove that the elementary bound is essentially optimal: construct, for every sufficiently small $s>0$ (or for a sequence $s_k\to 0$), a set $A_s\subseteq\mathbb{R}^n$ with $\dim_{\mathrm{P}}A_s\le s$ and an $A_s$-restricted Kakeya set $K_{A_s}$ with $\dim_{\mathrm{H}}K_{A_s}\le n-s+o(s)$ as $s\to 0$.

The statement must not be narrowed: the bound in form 1 must hold for all $A$ with $\dim_{\mathrm{P}}A\le s$ (not merely regular or structured $A$), and the construction in form 2 must supply segments in all directions $e\in S^{n-1}$. Progress confined to the regime where the bush-lifted or unrestricted Kakeya bounds already dominate (in $\mathbb{R}^4$, roughly $s\ge 1/2$) does not by itself resolve the question.

References

1. Jonathan M. Fraser and Lijian Yang, "Hausdorff dimension of restricted Kakeya sets", arXiv:2505.05709 (2025), DOI 10.48550/arXiv.2505.05709, https://arxiv.org/abs/2505.05709
2. Hong Wang and Joshua Zahl, "Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions", arXiv:2502.17655 (2025), https://arxiv.org/abs/2502.17655
3. Nets Hawk Katz and Joshua Zahl, "A Kakeya maximal function estimate in four dimensions using planebrushes", Revista Matemática Iberoamericana 37 (2021), no. 1, 317–359, DOI 10.4171/rmi/1219, https://doi.org/10.4171/rmi/1219
4. Jonathan Hickman, Keith M. Rogers, and Ruixiang Zhang, "Improved bounds for the Kakeya maximal conjecture in higher dimensions", American Journal of Mathematics 144 (2022), no. 6, 1511–1560, DOI 10.1353/ajm.2022.0037, https://doi.org/10.1353/ajm.2022.0037
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8. Joshua Zahl, "A Survey of the Kakeya conjecture, 2000–2025", arXiv:2512.09397 (2025), https://arxiv.org/abs/2512.09397
9. Jonathan M. Fraser and Lijian Yang, "Fourier analytic variants of the Furstenberg and Kakeya problems", arXiv:2605.21668 (2026), https://arxiv.org/abs/2605.21668
10. Anthony Gauvan, "Restricting directions for Kakeya sets", arXiv:2204.01408 (2022), https://arxiv.org/abs/2204.01408
11. Tamás Keleti and András Máthé, "Equivalences between different forms of the Kakeya conjecture and duality of Hausdorff and packing dimensions for additive complements", arXiv:2203.15731 (2022), https://arxiv.org/abs/2203.15731
12. Cheng Chen, Jianbin Li, Longhui Li, Bochen Liu, Ruichen Sun, Haoran Wang, and Saijie Zhou, "Some constructions of restricted Kakeya sets", arXiv:2609.06556 (2026), DOI 10.48550/arXiv.2609.06556, https://arxiv.org/abs/2609.06556