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 ...



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


