Improved bounds for the Kakeya maximal conjecture in higher dimensions

Jonathan Hickman; Keith M. Rogers; Ruixiang Zhang

Improved bounds for the Kakeya maximal conjecture in higher dimensions

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Pith's one-line read This paper proves the Kakeya maximal conjecture for all p ≥ 1 + min₂₌ₖ₌ₙ max{2n/((n−1)n+(k−1)k), 1/(n−k+1)}, improving the known range in dimensions 5 and all n ≥ 7.

desk verdict Genuine first progress on the Kakeya maximal conjecture in n=5 and n≥7, with a credible new multiscale Wolff-axiom proof; worth serious refereeing. read the letter →

arxiv 1908.05589 v1 pith:NGT32EV7 submitted 2019-08-14 math.CA math.MG

Jonathan Hickman , Keith M. Rogers , Ruixiang Zhang This is my paper · ORCID

classification math.CAmath.MG MSC 42B2528A7814P10

keywords KakeyamaximalconjecturesetpolynomialpartitioningmultiscaleWolffaxiomsbroadnormssemialgebraicvolumeboundsHausdorffdimensionharmonicanalysis

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 tries to establish that the Kakeya maximal conjecture holds for a wider range of Lebesgue exponents than was previously known, specifically for all $p \geqslant 1 + \min_{2 \leq k \leq n} \max\{2n/((n-1)n+(k-1)k), 1/(n-k+1)\}$, improving the known range in five dimensions and in every dimension $n \geqslant 7$. The proof adapts the polynomial partitioning method for Fourier restriction to the Kakeya problem by writing the induction argument as an explicit recursive algorithm, which brings out additional multiscale geometric information about direction-separated tubes. The authors show that such tubes satisfy a multiscale version of the polynomial Wolff axioms, with the proof resting on a new volume bound for certain semialgebraic sets formed by nested families of line segments. If the main estimate is correct, it also yields improved Hausdorff dimension lower bounds for Kakeya sets in an infinite sequence of dimensions, of the form $\dim_H K \geqslant (2-\sqrt{2})n + 3/2 - 1/\sqrt{2} - \varepsilon$.

What carries the argument

The load-bearing object is the multiscale polynomial Wolff axiom (Theorem 1.4): for direction-separated $\delta$-tubes and nested varieties $Z_k,\dots,Z_m$ of dimensions $k,\dots,m$ with nested balls $B_{\lambda_k} \subseteq \cdots \subseteq B_{\lambda_m}$, the number of tubes $T$ satisfying $|T \cap B_{\lambda_j} \cap N_\rho Z_j| \geqslant \lambda_j |T|$ for all $j = k,\dots,m$ is bounded by $C_{n,d,\varepsilon}(\prod_{j=k}^{m-1} \rho/\lambda_j)(\rho/\lambda_m)^{n-m}\delta^{-(n-1)-\varepsilon}$. This estimate is sharp up to the $\delta^{-\varepsilon}$ factor, as nested planes show. The proof reduces to a new volume bound (Lemma 3.5) for the semialgebraic set $S_m(I_m,\rho)$, the set of line segments that stay inside the $\rho$-neighbourhood of all the varieties $Z_j$ over a common time interval $I_m$. That volume bound is proved by a two-step induction: a trigonometric estimate compares $S_{\ell+1}$ with $S_\ell$ at a slightly larger radius, and an algebraic estimate, built from polynomial parametrizations and degree-counting from algebraic geometry, governs how these sets expand over a longer dyadic interval.

What would settle it

Compute $|S_m(I_m,\rho)|$ for a small explicit configuration, for instance $n=4$, $m=3$, $k=2$, with $Z_2$ a plane, $Z_3$ a quadric surface, nested unit balls, and dyadic intervals $I_2, I_3$ chosen so that $\rho/\lambda_2$ and $\rho/\lambda_3$ are small but unequal, and compare the measured volume with the right-hand side of Lemma 3.5. Exceeding the claimed bound by more than the admissible $\delta^{-\varepsilon}$ factor would refute the lemma and the main theorem.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: for every dimension $n \geqslant 2$, the Kakeya maximal inequality $(K_p)$ holds for $p \geqslant 1 + \min_{2 \leq k \leq n} \max\{2n/((n-1)n + (k-1)k), 1/(n-k+1)\}$. This yields the uniform bound $p \geqslant 1 + (2-\sqrt{2})^{-1}(n-1)^{-1}$ in all dimensions, which is strictly stronger than the previous best range, and gives new endpoints in low dimensions, such as $p = 18/13$ for $n=5$ and $p = 34/27$ for $n=7$. The proof proceeds by first proving a k-broad estimate (Theorem 4.1) for the Kakeya maximal function, then converting it to the usual linear maximal estimate via a broad-to-linear reduction. The broad estimate is obtained by feeding a recursive polynomial-partitioning algorithm into a new multiscale polynomial Wolff axiom (Theorem 1.4), which controls the number of tubes that have a substantial intersection with a whole nested chain of algebraic varieties at several scales.

Load-bearing premise

The entire argument rests on Lemma 3.5, the new volume bound for the semialgebraic sets $S_m(I_m,\rho)$: if that bound fails at any of the nested scales and varieties produced by the recursive algorithm, then the multiscale polynomial Wolff axioms and hence the main maximal estimate no longer follow.

Editorial extensions

If this is right

The Kakeya maximal conjecture is now known for $p \geqslant 1 + (2-\sqrt{2})^{-1}(n-1)^{-1}$ in every dimension, with strictly better exponents in dimensions 5 and all $n \geqslant 7$.
Every Kakeya set in $\mathbb{R}^n$ has Hausdorff dimension at least $(2-\sqrt{2})n + 3/2 - 1/\sqrt{2} - \varepsilon$ for an infinite sequence of dimensions (Corollary 9.1).
The same multiscale Wolff axioms give an improved range for the polynomial Wolff axiom conjecture, namely $p \geqslant 1 + \min_{2 \leq k \leq n} \max\{(n/(n-1))^{n-k}, (n-1)/(n-k+1)\}(n-1)^{-1}$ (Theorem 9.5).
The k-broad estimates hold for $p \geqslant 1 + 2n/((n-1)n + (k-1)k)$, and each such estimate can be promoted to a genuine $L^p$ maximal inequality once $p$ is also at least $(n-k+2)/(n-k+1)$.

Reading between the lines

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

Because Theorem 1.4 is sharp up to $\delta^{-\varepsilon}$ (nested planes saturate it), the multiscale volume bound is likely the true bottleneck for further endpoint improvements; if Lemma 3.5 could be upgraded to a logarithmic loss, the $\varepsilon$ in Theorem 1.2 should disappear.
The recursive-algorithm formulation is essentially a way to optimize the trade-off between cellular and algebraic steps; one could program the algorithm to search for the best exponent range over all choices of the parameters $k, m, \gamma_j$, potentially extending the method to other transversality problems.
The same $(2-\sqrt{2})n$ asymptotics that appeared in the earlier sum-difference approach to Kakeya reappear here through a completely different route, suggesting that $2-\sqrt{2}$ may be a structural constant for this family of polynomial partitioning arguments rather than an artifact of one technique.
The multiscale Wolff axioms imply k-linear estimates through the known dominance of k-linear over k-broad norms, so one might expect new multilinear restriction or Kakeya estimates at nearby exponents in dimensions covered by Theorem 1.4.

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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

Two things you should know. First, this is a real advance: it is the first improvement over the Katz–Tao exponent in n=5 and all n≥7, and it comes with a new multiscale version of the polynomial Wolff axioms (Theorem 1.4) that is likely to be reused. Second, I read the proof looking for a reason to dismiss it, and I did not find one. The chain from Lemma 3.5 to Theorem 1.2 is long but coherent; the Wongkew-type volume bound is the delicate heart, and the two-step induction that proves it checks out. The recursive reformulation of Guth's induction is a genuinely new idea, not a cosmetic rewrite.

The paper is honest: Remark 1.5 acknowledges Zahl's independent simultaneous work on the same maximal estimates. That does not reduce the value here; the multiscale Wolff axioms are independently established and useful.

The soft spots are real but not load-bearing. Proposition 5.7 and Theorem 9.5 are stated without proof. They are side results and do not feed into Theorem 1.2, but in a paper this long the authors should either provide the arguments or give precise references. Also the proof is not machine-checked, so the usual medium risk of a hidden algebraic slip remains; the stress-test note raised a possible 'iff' overstatement in (17) and I agree it is only an overstatement, since the covering uses the forward direction plus Corollary A.3. No gap.

Citation pattern is fine: the paper leans on Katz–Rogers and Hickman–Rogers for the single-scale case and builds outward; it does not assume the target result.

Bottom line: this is a serious paper for people working in harmonic analysis, geometric measure theory, and additive combinatorics. It deserves a careful referee, not a desk reject. I would bring it to the reading group and cite it. My recommendation is to accept with minor revision: fix the unproved side claims, either by proof or reference, and keep the presentation of Lemma 3.5 as is, maybe with a little more room for the induction.

Referee Report

0 major / 5 minor

Summary. The paper proves the Kakeya maximal conjecture in R^n for p at least 1 + min_{2<=k<=n} max{2n/((n-1)n+(k-1)k), 1/(n-k+1)} (Theorem 1.2), improving the previous best ranges in n=5 and all n>=7. The proof adapts Guth's polynomial-partitioning argument for the restriction problem to Kakeya, writing the induction as a recursive algorithm and introducing a multiscale version of the polynomial Wolff axioms (Theorem 1.4). The key new tool is a Wongkew-type volume bound for the semialgebraic sets S_m(I_m, rho) (Lemma 3.5), proved by a two-step induction mixing trigonometric and algebraic estimates. Theorem 1.2 follows from a k-broad estimate (Theorem 4.1) together with the Bourgain--Guth induction mechanism (Proposition 4.2). The paper also derives Hausdorff dimension bounds for Kakeya sets in an infinite sequence of dimensions (Corollary 9.1) and discusses variants such as the Guth--Zahl polynomial Wolff conjecture (Theorem 9.5).

Significance. If correct, this is a genuine advance on a central open problem in harmonic analysis, improving the Katz--Tao range in all sufficiently high dimensions under consideration and in dimension 5. The main technical novelty, Lemma 3.5, appears to be a substantial and plausible multiscale extension of Wongkew's theorem, and the overall proof is carefully structured: Theorem 1.2 is reduced to Theorem 4.1 and Proposition 4.2, Theorem 4.1 is proved via the recursive algorithms and the structural estimate in Section 8, and Theorem 1.4 is proved from Lemma 3.5. The argument is not circular: the target Kakeya maximal bounds are not assumed, and the proof builds on earlier single-scale polynomial Wolff axioms of Katz--Rogers and the Bourgain--Guth mechanism. The paper also gives explicit numerology, tables of exponents, and an honest discussion of the limitations and of the independent simultaneous work of Zahl (Remark 1.5). For these reasons the paper merits publication, provided the intricate proof of Lemma 3.5 is accepted as correct; I did not identify a concrete load-bearing error.

minor comments (5)

[Section 3.2, Lemma 3.7] The 'if and only if' assertion in (17) is stronger than what the subsequent argument needs; the proof only uses the implication from membership in L_ell(rho,t_ell) to containment of the line segment in S_ell(J,rho). The reverse implication should either be justified or the statement should be weakened accordingly.
[Section 3.2, proof of Lemma 3.5] The displayed inequality following the dyadic decomposition is easy to misread: the sum over J with the condition |S_ell(J,rho)| >= 4|J|rho^{n-1} appearing in the subscript should be typeset as a sum over {J in J : |S_ell(J,rho)| >= 4|J|rho^{n-1}} to avoid confusion with a product of factors.
[Section 5, Proposition 5.7] Proposition 5.7 is stated without proof. It is not used in the proof of Theorem 1.2, but as a proposition in the text it should either carry a proof or be explicitly labelled as a remark with an omitted proof.
[Section 9.3, Theorem 9.5] Theorem 9.5 is presented as a theorem but no proof is supplied. Since it is not needed for the main Kakeya maximal theorem, this is not a blocking issue, but the text should clearly state that the proof is omitted or that the statement is a consequence of a variant of the earlier argument.
[Section 3 and Section 4] There are minor typographical errors: 'semiaglebraic' appears in the proof of Theorem 1.4 and 'algbraic' appears in the overview of Section 4; both should be corrected.

Circularity Check

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No significant circularity: the Kakeya maximal bound is derived from the independently proven multiscale Wolff axioms (Theorem 1.4), whose proof rests on the new Wongkew-type volume lemma (Lemma 3.5); the target estimates are never assumed.

full rationale

The paper's derivation chain is not circular. The main Kakeya maximal estimate (Theorem 1.2) is obtained by combining the k-broad estimate (Theorem 4.1) with the Bourgain-Guth mechanism (Proposition 4.2), and Theorem 4.1 is proven via the polynomial partitioning algorithm culminating in the structural second key estimate in Section 8. The only genuinely new ingredient in this chain is the multiscale polynomial Wolff axioms, Theorem 1.4, which is proven in Section 3 from the Wongkew-type volume bound Lemma 3.5. Lemma 3.5 concerns the Lebesgue measure of concrete semialgebraic sets Sm(Im, rho) formed by unions of line segments constrained by nested neighbourhoods of algebraic varieties; its proof is an induction that combines Wongkew's theorem, the Tarski-Seidenberg theorem, Gromov's algebraic lemma, and elementary trigonometric and algebraic estimates. None of these inputs assumes the Kakeya maximal conjecture, any k-broad estimate, or the final theorem. In particular, the cardinality estimate in Theorem 1.4 is applied in Section 8 only after it has been established, and the volume lemma used to prove it is independent of the Kakeya problem. The paper does cite the authors' own prior work: Katz-Rogers [25] for the single-scale polynomial Wolff axioms and the semialgebraic section lemmas, and Hickman-Rogers [23] for the recursive algorithm. These are prior, independently stated results whose assumptions do not include the target Kakeya estimates; moreover, the recursive algorithm is reformulated and its properties are proved in the present paper rather than assumed. The external benchmarks (Bourgain-Guth Proposition 4.2, Guth's polynomial partitioning Theorem 6.1 and Lemma 7.1, Wongkew's theorem, Gromov's lemma) are all standard or previously established results not derived from the paper's conclusion. Remark 1.5, noting independent simultaneous work by J. Zahl, further supports that the main result is not an artifact of a self-citation loop. No fitted parameters are renamed as predictions, no target estimate is imported by definition, and no uniqueness theorem is invoked to force the choice of argument. The unresolved items mentioned in the text, such as the omitted proof of Proposition 5.7 and the unproved Theorem 9.5, are explicitly non-load-bearing for Theorem 1.2. Therefore, although the proof is intricate and not machine-checked, there is no identifiable circular step.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The proof rests on standard external theorems rather than on new postulated entities or fitted constants. The central new ingredients, the Wongkew-type volume bound (Lemma 3.5) and the recursive-algorithm decomposition, are proven in the text. No numerical parameters are fitted to data; all constants are chosen to close epsilon-delta estimates. The main external inputs are Wongkew's tubular neighbourhood theorem, the Tarski-Seidenberg projection theorem, Gromov's algebraic lemma, Guth's polynomial partitioning theorem, and the single-scale polynomial Wolff axioms of Guth, Zahl, and Katz-Rogers. None of these assume the Kakeya maximal conjecture, so the argument is not circular.

assumptions (7)

standard math Wongkew's theorem (Theorem A.1) bounds the volume of tubular neighbourhoods of algebraic varieties.
Used in Lemma 3.5 and in the proof of Theorem 1.4 to bound volumes of N_rho Z_j; quoted from [43] in Appendix A.
standard math Tarski-Seidenberg projection theorem (Theorem A.2) and its semialgebraic section corollary (Corollary A.3).
Used to form semialgebraic sections and ensure complexity bounds in the proof of Lemma 3.7 and Theorem 1.4; see Appendix A.
standard math Gromov's algebraic lemma (Lemma A.4) provides C^r parametrizations of compact semialgebraic sets.
Used to parametrize line families in Lemma 3.7 and Section 3.3; quoted from [33,8].
domain assumption Guth's polynomial partitioning theorem (Theorem 6.1) for L^1 functions, including cellular and algebraic cases.
Borrowed from Guth [19]; the paper relies on it to drive the recursive algorithm in Section 7. This is a known result, not proven in this paper.
domain assumption The polynomial Wolff axioms in the single-scale form (Theorem 1.3) from [18,44,25].
Used as the base case and as motivation for the multiscale generalization in Theorem 1.4; attributed to Guth, Zahl, and Katz-Rogers.
domain assumption Katz-Tao sum-difference bounds [27] and Bourgain's bound [6] as the prior state-of-the-art benchmarks.
Used in the introduction (Figure 1) to compare the new ranges; not used as input to the proof.
standard math Equidistribution of (sqrt(2)-1)n modulo 1.
Used in Section 9.1 to show the exponent is attained for an infinite sequence of dimensions; standard result from the theory of equidistribution.

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Pith. "Pith review of Improved bounds for the Kakeya maximal conjecture in higher dimensions." pith.science (2026). https://pith.science/paper/NGT32EV7

@misc{pith2026190805589,
  author       = {Pith},
  title        = {Pith review of: Improved bounds for the Kakeya maximal conjecture in higher dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NGT32EV7}},
  note         = {Machine review of arXiv:1908.05589}
}

abstract

We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made available. To take advantage of this, we prove that direction-separated tubes satisfy a multiscale version of the polynomial Wolff axioms. Altogether, this yields improved bounds for the Kakeya maximal conjecture in $\mathbb{R}^n$ with $n=5$ or $n\ge 7$ and improved bounds for the Kakeya set conjecture for an infinite sequence of dimensions.

Figures

Figures reproduced from arXiv: 1908.05589 by the authors.

**Figure 1.** The state-of-the-art for the Kakeya maximal conjecture in low dimensions. New results are highlighted. In 1999, Bourgain [6] improved the state-of-the-art in higher dimensions using sum-difference theory from additive combinatorics. This technique was refined by Katz and Tao [26, 27, 28], proving that Conjecture 1.1 is true in the range p > 1 + 7 4 1 n−1 . The purpose of the present article is to extend this range u… view at source ↗

**Figure 2.** The set Sm(J, ρ) is formed by a union of line segments la,d(J) which have the property that la,d(Ij ) ⊆ NρZj ∩ Bλj for k 6 j 6 m. Given any interval J ⊆ R, we define Sm(J, ρ) := \m j=k la,d(t) : t ∈ J, (a, d) ∈ [−1, 1]2(n−1), la,d(Ij ) ⊆ NρZj ∩ Bλj [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** The trigonometric argument. Thus, by (13), altogether we find that |Sℓ(J, ρ) ∩ NρZℓ+1| 6 X B∈B |B ∩ NρZℓ+1| .d ρ|J| |Iℓ| ℓ+1 ρ n−(ℓ+1) |Iℓ| ρ|J| n |Sℓ(J, 2ρ)|, as desired. It remains to verify the claim. Letting rℓ := ρ|J|/(4nd|Iℓ|), by an elementary covering argument it suffices to show that NrℓSℓ(J, ρ) ∩ [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Forming a semialgebraic section of the lines. Roughly speaking, the slice Sℓ(J, ρ)tℓ (shown as a blue vertical line above) is parametrised by a polynomial mapping F : R n−1 → R n−1 . We can find another polynomial mapping G: R n−1 → R n−1 which “selects” a single line …

**Figure 5.** Figure 5: The state-of-the-art for the Kakeya set conjecture in low dimensions. New results are highlighted. Provided ε > 0 is sufficiently small, this bound is stronger than that obtained by Katz–Tao [27]. On the other hand, the Hausdorff dimension bound provided by Theorem 1.2…

**Figure 6.** Figure 6: The current state-of-the-art for Conjecture 9.4 in low dimensions. Here the omega constant Ω ∈ (1/2, 1) is the solution to e Ω = Ω−1 . In particular, Theorem 9.5 implies that Conjecture 9.4 is true in the range p > 1 + Ω −1 n−1 , yielding an improvement over Wolff’s bo…

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