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https://arxiv.org/html/2608.22209

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The restriction, Bochner–Riesz, and local smoothing conjectures each imply the Kakeya conjecture.
Supports. An expository article by a leading harmonic analyst for the ICM proceedings. It states the implications as known results, explains the mechanism (wave-packet decompositions that compress into Kakeya-like configurations, in the manner of Fefferman's 1971 disk multiplier counterexample), and points to surveys for the proofs. It also adds a fourth conjecture, Montgomery's in analytic number theory, to the list of conjectures that imply Kakeya. Sep 15, 2026

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Text185 passages · 1 marked

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1 The Kakeya problem1 passage

3 The road to the solution of the Kakeya conjecture0 passages

4 Further directions0 passages

5 Acknowledgments0 passages

References0 passages

License: arXiv.org perpetual non-exclusive license2 passages

Terence Tao1 passage

Email address: tao@math.ucla.edu0 passages

Date: August 23, 20264 passages

2020 Mathematics Subject Classification0 passages

Primary 28A75; Secondary 11M06, 28A78, 42B15, 42B25, 42B370 passages

1. The Kakeya problem78 passages · 1 marked

In the decades that followed, many further connections of this type were unearthed, in which the objects being analyzed contained “wave packets” that could point in different directions due to some “curvature” in the problem could end up being compressed into a Kakeya-like configuration, thus leading to unwanted behavior. Because of this, many long-standing conjectures in harmonic analysis, number theory, and partial differential equations were shown to imply the Kakeya conjecture, in that any counterexample to the Kakeya conjecture could be converted to a counterexample to these latter conjectures. Some examples include:

The restriction, Bochner–Riesz, and local smoothing conjectures each imply the Kakeya conjecture.VerifiedReading · this claim’s Steward

Supports. An expository article by a leading harmonic analyst for the ICM proceedings. It states the implications as known results, explains the mechanism (wave-packet decompositions that compress into Kakeya-like configurations, in the manner of Fefferman's 1971 disk multiplier counterexample), and points to surveys for the proofs. It also adds a fourth conjecture, Montgomery's in analytic number theory, to the list of conjectures that imply Kakeya.

3. The road to the solution of the Kakeya conjecture22 passages

4. Further directions1 passage

5. Acknowledgments1 passage

References60 references

Instructions for reporting errors7 passages

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Link: https://arxiv.org/html/2608.22209
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Stored text: 43,085 characters, retrieved Sep 15, 2026 · 23 sections, 185 passages
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Nov 9, 2026DividedSegmenter · Divided into 208 parts by its own structure
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