Arithmetic Kakeya conjecture

Establish the arithmetic Kakeya conjecture by proving that the infimum of the sum-difference exponents \((R;-1)\), taken over all finite sets of rational slopes \(R\), equals \(1\), thereby implying that Kakeya and Nikodym sets in every dimension have full Minkowski and Hausdorff dimension.

Background

The paper defines (R;s)(R;s) as the least exponent controlling the entropy of the projection at slope ss in terms of the projections at slopes in a finite set RR. Improvements below the trivial upper bound of $2$ yield dimension bounds for Kakeya and Nikodym sets.

The arithmetic Kakeya conjecture predicts that, as the slope set becomes sufficiently rich, these exponents can approach $1$. The paper records that the conjecture is known only in dimensions at most three and that the best currently known upper bound for the relevant infimum is 1.67513…1.67513\ldots, so the conjecture remains unresolved.

References

In particular, if one can establish the arithmetic Kakeya conjecture \begin{equation}\label{infr} \inf_R (R;-1) = 1 \end{equation} then this would imply that Kakeya and Nikodym sets in $Rd$ have full Minkowski and Hausdorff dimension for all $d$.

infr:

inf⁡R(R;−1)=1\inf_R (R;-1) = 1

— Sum-difference exponents for boundedly many slopes, and rational complexity  (2511.15135 - Tao, 19 Nov 2025) in Section 1.1, “The arithmetic Kakeya conjecture”