Hausdorff dimension of restricted Kakeya sets

Jonathan M. Fraser; Lijian Yang

REVIEW 3 major objections 5 minor 14 references

Hausdorff dimension of restricted Kakeya sets

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Restricting the midpoints of the unit segments in a Kakeya set to a set of dimension at most $s$ forces the set to have Hausdorff dimension at least $n-s$, and the bush argument improves this to $n-g_n(s)$.

desk verdict A new restricted Kakeya framework with a clean elementary bound and a plausible bush-argument improvement, held up mainly by one imported parallelogram estimate that needs a proof or a precise reference. read the letter →

arxiv 2505.05709 v2 pith:MYVVFMOB submitted 2025-05-09 math.CA

Jonathan M. Fraser , Lijian Yang This is my paper · ORCID

classification math.CA MSC 28A8028A78

keywords KakeyaconjecturesetrestrictedmaximalfunctionHausdorffdimensionpackingboxbushargument

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies Kakeya sets whose defining unit segments are forced to pass through a prescribed set $A$: for every direction $e$ there is a unit segment in direction $e$ whose midpoint lies in $A$. If $A$ has packing dimension at most $s$, the authors prove that any such $A$-restricted Kakeya set in $\mathbb{R}^n$ has Hausdorff dimension at least $n-s$; the simple argument is that $K-A$ contains a ball and the product inequality transfers the dimension. Their main theorem converts a Kakeya maximal estimate in dimension $n-1$ into a weak-type estimate for the $A$-restricted maximal function in $\mathbb{R}^n$ via the bush argument, yielding the improved bound $n-g_n(s)$ for a function $g_n$ built from the lower-dimensional exponent. In $\mathbb{R}^4$ this gives $\max\{19/5 - 3s/5, 4-s\}$, and in higher dimensions it gives explicit bounds from known maximal estimates. The same lower bounds hold if $A$ only contains an arbitrary point on each segment rather than the midpoint, and they force full dimension $n$ when such a set has very small packing dimension.

What carries the argument

The central object is the $A$-restricted Kakeya maximal function $K_{\delta,A}(f)(e)=\sup_{a\in A}|T_\delta^e(a)|^{-1}\int_{T_\delta^e(a)}|f|$, where $T_\delta^e(a)$ is the $\delta$-neighbourhood of the unit segment in direction $e$ with midpoint $a$. Weak-type estimates for this operator are converted into Hausdorff dimension lower bounds by Lemma 2.3. The improved estimates are carried by the bush argument: iteratively find a point where many $\delta$-separated tubes overlap, remove that bush, and control the remaining directions with an $n$-dimensional parallelogram maximal estimate that is imported from the lower-dimensional Kakeya maximal hypothesis. The elementary $n-s$ bound is carried instead by the fact that $K-A$ contains a ball of radius $1/2$, together with product dimension inequalities.

What would settle it

Construct, for some $s$ in the range where the improvement is claimed, an $A$-restricted Kakeya set in $\mathbb{R}^4$ with $\dim_B A\le s$ but $\dim_H K_A<19/5-3s/5$; equivalently, exhibit functions supported in annuli $B(0,2r)\setminus B(0,r)$ for which the parallelogram maximal estimate (3.22) fails on $\mathbb{R}^4$ at the stated exponents.

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Extended reading notes

Core claim

The central claim is Theorem 2.6: if the Kakeya maximal function in $\mathbb{R}^{n-1}$ satisfies an estimate of the form $\|(f)^*_\delta\|_{L^{p_{n-1}}(S^{n-2})} \lesssim_\varepsilon \delta^{-h_{n-1}-\varepsilon}\|f\|_{L^{p_{n-1}}}$, then for any $A$ with upper box dimension at most $s$, the $A$-restricted Kakeya maximal function in $\mathbb{R}^n$ satisfies a weak-type $L^p$ bound with $p=(p_{n-1}+n(p_{n-1}-1)+1)/p_{n-1}$ and exponent $\beta=(h_{n-1}p_{n-1}+sp_{n-1}-s)/(p_{n-1}+n(p_{n-1}-1)+1)$. A reduction lemma converts this into the Hausdorff dimension lower bound $n-g_n(s)$, where $g_n(s)=h_{n-1}+s-s/p_{n-1}$; the elementary bound $n-s$ always holds, so the combined lower bound is $\max\{n-s,n-g_n(s)\}$. The paper records packing-dimension versions and shows the same bounds hold when $A$ is merely met by every segment at some point, and it proves matching statements for the restricted Kakeya maximal function.

Load-bearing premise

The improved bound in dimensions $n\ge 4$ rests on an $n$-dimensional parallelogram maximal estimate imported from an earlier work whose explicit case is only three-dimensional; if that estimate fails, the stronger $n-g_n(s)$ bound is not established.

Editorial extensions

If this is right

In $\mathbb{R}^4$, a Kakeya set whose segment midpoints lie in a set of upper box dimension $s$ has Hausdorff dimension at least $\max\{19/5-3s/5,4-s\}$, which exceeds the general four-dimensional lower bound for a range of $s$.
The same dimension bounds hold when the prescribed set contains an arbitrary point of each segment, not necessarily its midpoint (Corollary 2.13).
If a Kakeya set contains a set $P$ of packing dimension less than $\varepsilon$ meeting every direction's segment, then the set has Hausdorff dimension $n$ (Corollary 2.14).
Any future improvement of the Kakeya maximal estimate in dimension $n-1$ feeds through Theorem 2.6 to improve the restricted dimension bound in $\mathbb{R}^n$.
The corresponding estimates for the $A$-restricted Kakeya maximal function hold as weak-type $L^p$ bounds, giving maximal-function analogues of each dimension statement.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The transfer principle suggests that the restricted problem is the quantitative core of the Kakeya conjecture: because the restriction can only help, any counterexample to a restricted bound would also be a counterexample to the unrestricted conjecture, so the new bounds show how much midpoint freedom is needed to break the trivial product bound.
The authors' remark that the improved bound does not beat $n-s$ for very small $s$ leaves open a possible second transition near $s=0$; testing this in $\mathbb{R}^4$ with near-extremal restricted sets around $s=1/2$ would clarify whether the bush bound is sharp.
The same bush and parallelogram machinery should extend to families of $k$-dimensional disks in place of unit segments, replacing the sphere of directions by a Grassmannian and using a base Kakeya estimate for $k$-planes, yielding analogous $n-s$ and bush-improved bounds.
In three dimensions, where the general Kakeya conjecture has been settled, the restricted problem is automatically solved; the meaningful new information from these bounds lies in four and higher dimensions, where the gap between the two bounds suggests where a sharper argument might begin.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper introduces a genuinely new restricted Kakeya problem: the segments must have their midpoints (or, later, any fixed point on each segment) in a prescribed set A of packing or box dimension at most s. The elementary bound dim_H K_A ≥ n−s is clean and correct, via both a pigeonhole/covering argument and a product argument. The genuinely new content is the bush-argument improvement, dim_H K_A ≥ n−g_n(s), built from Kakeya maximal estimates in one lower dimension. For n=4 this yields max{19/5 − 3s/5, 4−s} on part of the range, beating current general Kakeya bounds there, and the same mechanism operates in higher dimensions using Hickman–Rogers–Zhang. The proofs are mostly transparent and the maximal-function analogues are a nice bonus.

The soft spots are real but local. First, the piecewise formula in Section 2.2 mis-states the s≥3 branch: maximizing over p gives 2, not 19/5−3s/5, because the p=1 term dominates there. The same type of issue appears in the general piecewise corollary; the saturated regime should be a constant 2, not a decreasing line. This does not damage the theorems but the examples need correcting. Second, Corollary 2.12 claims the full-direction results extend from upper box dimension to packing dimension. The proof covers A by pieces, selects one piece whose direction set has positive measure, rotates that direction set to fill most of the sphere, and then applies the full-direction corollaries to the rotated union. The corollaries as stated require segments in every direction. The proof sketch in Corollary 2.13 says the arguments go through for positive-measure direction sets, but that extension is not actually proved. A short lemma would fix this. Third, and most importantly, the improved bound depends on Lemma 3.5, a parallelogram maximal estimate imported from Bourgain. The authors note that only the n=3 case is explicit in [B91, Lemma 1.52] and that the n-dimensional version is only implicit. Since Theorem 2.6 uses this lemma at exactly one step, the n−g_n(s) bound for n≥4 is only as solid as that citation. If the estimate is not literally in Bourgain, the proof needs to supply it. This is the one thing I would require before publication.

The citations and attributions are otherwise appropriate, and the paper is honest about what is new and what is not. Researchers working on Kakeya-type problems, restricted maximal functions, or dimension theory of Besicovitch sets will find it useful. The central idea is sound and the flaws are addressable. I would send this to a serious referee rather than desk-reject it, and expect a revision that corrects the example formulas and tightens the two proof gaps.

Referee Report

3 major / 5 minor

Summary. The paper studies restricted Kakeya sets in R^n: compact sets containing a unit line segment in every direction whose midpoints are constrained to lie in a prescribed set A. The main results give lower bounds for the Hausdorff dimension of such sets in terms of the upper box or packing dimension s of A. The elementary bound dim_H K_A ≥ n−s is proved by a covering/pigeonhole argument and a product argument. The main new contribution is an improved bound dim_H K_A ≥ n−g_n(s) obtained by adapting Bourgain's bush argument to the restricted setting, where g_n(s) is built from known Kakeya maximal estimates in dimension n−1. Explicit consequences are worked out for R^3, R^4, and R^{10}, and maximal-function analogues of the dimension statements are given. A final section extends the statements from midpoints to arbitrary selected points on each segment and from upper box dimension to packing dimension.

Significance. If the improved bounds are correct, the paper provides a genuinely new family of dimension estimates for restricted Kakeya sets, including an explicit bound in R^4 that beats both the trivial n−s bound and the current general Kakeya lower bound on part of the parameter range. The elementary Proposition A is clean and the maximal-function framework is well chosen. The paper is honest about its external inputs: it builds on Wolff, Hickman–Rogers–Zhang, and Bourgain rather than claiming a proof of the Kakeya conjecture. However, the central improved bound rests on an imported n-dimensional parallelogram maximal estimate that is not proved here, and the packing-dimension upgrade contains a logical gap. These issues are load-bearing, so the paper is not ready in its present form.

major comments (3)

[Section 3.3, Lemma 3.5 / Eq. (3.22)] The n-dimensional parallelogram maximal estimate is imported from [B91] without proof, and the text itself states that only the n=3 case is explicit in [B91, Lemma 1.52], with the extension to all n asserted as implicit in [B91, p.158, (2.8)]. This estimate is the only route by which the bush iteration in Theorem 2.6 produces the improved exponent n−g_n(s); without it, Corollary 2.7 and all subsequent improved bounds, including the n=4 example, are unsupported. Please provide a complete proof of (3.22) or a precise reference containing the full statement with proof for all n≥4.
[Section 2.4, proof of Corollary 2.12] The proof constructs K0 whose line segments cover only the direction set E0=∪ r_i(E_k), which has measure >1/2, but then applies Corollaries 2.5, 2.7, and 2.11, which require an A-restricted Kakeya set in the sense of Definition 2.1, i.e. a segment in every direction of S^{n-1}. Since K0 does not satisfy that definition, the application is invalid. The argument needs a genuinely proved positive-measure-direction version of the main theorems; as written, the packing-dimension upgrade and Corollary 2.13 are not established.
[Section 2.2, n=4 piecewise formula] The displayed formula after Corollary 2.7 misstates the branch for 3≤s≤4. For s≥3, the maximum of 4−(3−s)/p−(s−1) over 1≤p≤5/2 is attained at p=1 and equals 2, not 19/5−3s/5. The piecewise expression should have a separate branch 2 for 3≤s≤4, with the improved branch restricted to 1/2≤s<3. This error affects the stated theorem and Figure 1.

minor comments (5)

[References] Reference [C77] is misprinted: the American Journal of Mathematics entry gives volume 9, (2023), pages 1–22; it should be volume 99 (1977), pages 1–22.
[Section 3.3, proof of Theorem 2.6] The symbol E0 is used both for the initial measurable set E and for a 10δ/λ-separated subset of D0; the resulting notational collision is confusing and should be resolved.
[Corollary 2.13] The proof sketch says all arguments go through for endpoints and for positive-measure direction sets, but no formal verification is supplied; since this extension is also needed to repair Corollary 2.12, it should be proved explicitly or the statement should be marked conditional.
[Proposition 3.2] The hypothesis is stated with a universal quantifier over ε>0 in the preceding display, but the proposition text says 'for some ε>0'; the mismatch should be corrected.
[Introduction/Abstract] The abstract and Definition 2.1 discuss midpoints, while Corollary 2.13 later allows arbitrary selected points on each segment; the introduction would benefit from an early sentence signalling this intended generalization.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the restricted Kakeya bounds are derived from external maximal-function estimates, with no parameter fitted to the conclusions and no load-bearing self-citation.

full rationale

The derivation chain is independent of the results it aims to prove. Proposition A and Theorem 2.4 obtain the elementary bounds n-s from the box dimension of A via a pigeonhole argument and a geometric tube argument, with no use of the target dimension bound as an input. Theorem 2.6 and Corollary 2.7 take as hypotheses external Kakeya maximal function estimates in R^{n-1}, such as those of Wolff and Hickman-Rogers-Zhang, and combine them with Bourgain's bush argument to produce restricted maximal estimates in R^n. The parameters p and beta in (2.4)-(2.5) are explicit algebraic functions of the assumed exterior exponents, not fitted constants. Lemma 3.5, the parallelogram maximal estimate, is quoted from Bourgain [B91]; even if its extension to all n is only implicit in the cited source, it is an external mathematical input rather than a self-citation or a definitional restatement of the conclusion. The paper's own results are not used to justify their assumptions, and no prediction is equivalent to any fitted input by construction. The concern that Lemma 3.5 may not be fully established for n>=4 is a correctness or verification issue, not circularity, and does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proofs rest on standard dimension theory and known lower-dimensional Kakeya maximal estimates. The only non-standard imported item is the n-dimensional parallelogram maximal lemma from Bourgain, which is cited with an asserted implicit extension to all dimensions. No ad-hoc assumptions or new physical or mathematical entities are introduced.

assumptions (5)

domain assumption Wolff's Kakeya maximal estimate in R^3
Cited from [W95]; used in Section 2.2 for the n=4 example, not proved in this paper.
domain assumption HRZ Kakeya maximal estimate in R^{n-1}
Cited from [HRZ22], used for n at least 5 in Theorem 2.10 and Corollary 2.11.
domain assumption Bourgain's parallelogram maximal function estimate in R^n (Lemma 3.5)
The proof of Theorem 2.6 depends on this lemma; the n>3 case is asserted as implicit in [B91] but not proved here.
standard math Product dimension inequality dim_H(K x A) <= dim_H K + dim_P A
Used in Proposition A(2); cited from [F14, M95].
standard math Lebesgue density theorem on S^{n-1}
Used in the proof of Corollary 2.12 to find a positive-density subset of directions.

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Pith. "Pith review of Hausdorff dimension of restricted Kakeya sets." pith.science (2026). https://pith.science/paper/MYVVFMOB

@misc{pith2026250505709,
  author       = {Pith},
  title        = {Pith review of: Hausdorff dimension of restricted Kakeya sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MYVVFMOB}},
  note         = {Machine review of arXiv:2505.05709}
}

abstract

A Kakeya set in $\mathbb{R}^n$ is a compact set that contains a unit line segment $I_e$ in each direction $e \in S^{n-1}$. The Kakeya conjecture states that any Kakeya set in $\mathbb{R}^n$ has Hausdorff dimension $n$. We consider a restricted case where the midpoint of each line segment $I_e$ must belong to a fixed set $A$ with packing dimension at most $s \in [0, n]$. In this case, we show that the Hausdorff dimension of the Kakeya set is at least $n - s$. Furthermore, using the "bush argument", we improve the lower bound to $\max \{ n - s, n - g_n(s)\}$, where $g_n(s)$ is defined inductively. For example, when $n = 4$, we prove that the Hausdorff dimension is at least $\max\{\frac{19}{5} - \frac{3}{5}s,4-s\}$. We also establish Kakeya maximal function analogues of these results.

Figures

Figures reproduced from arXiv: 2505.05709 by the authors.

**Figure 1.** Lower bound for dimHKA in R 4 under the assumption that dimBA ⩽ s. The blue bound comes from Corollary 2.5 and the black bound comes from Corollary 2.7. The best lower bound is the maximum of these two and is shown as a solid line. 2.3. Higher Dimensions. According to [M15, Proposition 22.6], we have the following discrete version of (2.3). Proposition 2.8. Let 1 < pn−1 < ∞, p ′ n−1 = pn−1 pn−1−1 , hn−1 > 0 and 0 < … view at source ↗

**Figure 2.** Lower bound for dimHKA in R 10 under the assumption that dimBA ⩽ s. The blue bound comes from Corollary 2.5 and the black bound comes from Corollary 2.7. The best lower bound is the maximum of these two and is shown as a solid line. Since [∞ i=1 Ei = S n−1 , there exists some Ek such that |Ek|n−1 > 0. By Lebesgue’s density theorem on S n−1 , there exist finitely many elements r1, . . . , rN in the orthogonal group … view at source ↗

**Figure 3.** Mδ tubes whose centres are in the ball B(yi0 , 1 3 δ). which implies m′ ≳ λ n−1 N Nδ . It follows from (3.5) that the sets {E ∩ T ′ k \B(y, cλ)} m′ k=1 are disjoint because diam(T ′ k ∩ T ′ s ) ⩽ bδ |e ′ k − e ′ s | ⩽ bδ bδ/(cλ) = cλ for any k ̸= s in 1, . . . , m′ . Therefore, |E|n ≳ λδn−1m′ ≳ λδn−1λ n−1 N Nδ ≳ Nλn δ n+s+ε−1 . Together with (3.3), we obtain (3.2). 3.3. Proof of Theorem 2.6. By checking a simple exa… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Construction of the bushes Bi . The next step is to show that this construction must terminate after a finite number of steps. Proposition 3.3. The construction process terminates before step m, where (3.16) m ⩽ 1 ε0 |E|n δ −s−ελ −n [PITH_FULL_IMAGE:figures/full_fig_…

**Figure 5.** Figure 5: Geometric Observation for (3.23). a0 is the midpoint of xi and aξ. Using (3.21), we estimate (3.24) λ ≲ mX−1 i=0 1 [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

**Figure 6.** Figure 6: Geometric Observation for (3.27). P Substituting the stop condition in Proposition 3.3 for m and the first part of (3.18) for m−1 i=0 |Bi |n , ε0λ pn−1 ≲ε m log 1 δ pn−1−1 log 1 Xδ k=0 (2k δ) −1 δ −pn−1hn−1−ε 2 k δ mX−1 i=0 |Bi |n ⩽ mpn−1−1 log 1 δ pn−1 δ −pn−1hn…

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Reference graph

Works this paper leans on

14 extracted references · 12 canonical work pages

[1]

Bourgain

J. Bourgain. Besicovitch Type Maximal Operators and Applications to Fourier Analysis, Geom. Func. Anal., 1 , (1991), 147--187

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[2]

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J. Hickman, K. Rogers and R. Zhang. Improved bounds for the Kakeya maximal conjecture in higher dimensions, Amer. J. Math , 144 , (2022), 1511--1560

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[9]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Bourgain

J. Bourgain. Besicovitch Type Maximal Operators and Applications to Fourier Analysis, Geom. Func. Anal., 1 , (1991), 147--187

work page 1991

[2] [2]

C\'ordoba

A. C\'ordoba. The Kakeya Maximal Function and the Spherical Summation Multipliers, Amer. J. Math. , 9 , (2023), 1--22

work page 2023

[3] [3]

R. Davies. Some remarks on the Kakeya problem, Math. Proc. Cambridge Philos. Soc., 69, (1971), 417--421

work page 1971

[4] [4]

K. J. Falconer. Fractal Geometry: Mathematical Foundations and Applications , John Wiley & Sons, Hoboken, NJ, 3rd. ed., (2014)

2014

[5] [5]

Hickman, K

J. Hickman, K. Rogers and R. Zhang. Improved bounds for the Kakeya maximal conjecture in higher dimensions, Amer. J. Math , 144 , (2022), 1511--1560

work page 2022

[6] [6]

N. Katz, L. aba and T. Tao. An improved bound on the Minkowski dimension of Besicovitch sets in R ^3 , Ann. Math., 152 , (2000), 383--446

work page 2000

[7] [7]

Katz and T

N. Katz and T. Tao. New bounds for Kakeya problems, J. Anal. Math , 87 , (2002), 231--263

work page 2002

[8] [8]

Katz and J

N. Katz and J. Zahl. An improved bound on the Hausdorff dimension of Besicovitch sets in R ^3 , J. Amer. Math. Soc. , 32 , (2019), 195--259

work page 2019