# A reference for this possibly well-known fact concerning the Kakeya conjecture?

I believe I have read or heard somewhere that the Kakeya conjecture would follow from appropriate lower bounds for the minimal size of a subset of $\{ 1 , \cdots , N\}$ which contains a translate of every k-term arithmetic progression contained in $\{ 1 , \cdots , N\}$.

This may be well-known to experts (what I'm not) and I have been unable to locate an appropriate reference ...

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Katz & Tao's survey article cites this (but I don't know if it contains your specific point): J. Bourgain: Harmonic Analysis and Combinatorics: How Much May They Contribute to Each Other?, in Mathematics: Frontiers and Perspectives, V. Arnold, M. Atiyah, P. Lax, B. Mazur, eds., AMS 2000. –  Joseph O'Rourke Jul 24 '13 at 16:27
I think you refer to the sentence "Indeed,the Kakeya conjecture can be reformulated in terms of arithmetic progressions, and this can be used to connect the Kakeya conjecture to several difficult conjectures in number theory such as the Montgomery conjectures for generic Dirichlet series. We will not discuss this connection here, but refer the reader to [4]" from Katz&Tao. But as far as I can see, the reference given describes the connection between Dirichlet polynomials and Kakeya, but not the reformulation of Kakeya in terms of arithmetic progressions. –  js21 Jul 24 '13 at 16:52
However, the quote above confirms that (a) such a reformulation in terms of arithmetic progressions actually exist and (b) it is "well known" to at least two experts (Katz&Tao ...). –  js21 Jul 24 '13 at 17:01

## 1 Answer

A precise formulation of this implication may be found in Section 4 of the article

T. Wolff, Recent work connected with the Kakeya problem. Prospects in mathematics (Princeton, NJ, 1996), 129–162, Amer. Math. Soc., Providence, RI, 1999.

which may be found online at http://www.csun.edu/~vcmth014/191dn2.ps‎ . The lower bound for unions of arithmetic progressions is the claim (*) appearing in Proposition 4.2 of that paper. As discussed there, this is closely connected to Bourgain's observation that Montgomery's conjecture on Dirichlet series implies the Kakeya conjecture (indeed it implies (*), which in turn implies Kakeya).

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For those allergic to postscript files, here is a link to the pdf of Wolff's article: amathe.web.elte.hu/modern/wolff_review.pdf –  Vidit Nanda Jul 24 '13 at 18:19
For an alternate / more detailed exposition, see proposition 6 of Ben Green's notes dpmms.cam.ac.uk/~bjg23/restriction_kakeya_phenomena/notes10.ps –  Mark Lewko Jul 24 '13 at 18:39
Thanks all ! What I was looking for is proposition 4.3 in Wolff's article. Ben Green's notes are particularly clear and well-written. –  js21 Jul 25 '13 at 10:37