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 ...

Mathematics: Frontiers and Perspectives, V. Arnold, M. Atiyah, P. Lax, B. Mazur, eds., AMS 2000. $\endgroup$