Arithmetic Kakeya conjecture (entropy formulation)

Establish that for any finite set of distinct slopes {r_1,…,r_k} and target slope r_∞, the entropy inequality constant C_arith({r_1,…,r_k}; r_∞) can be made arbitrarily close to 1 for discrete random variables on finite supports.

Background

The arithmetic Kakeya conjecture translates geometric Kakeya-type phenomena into discrete entropy inequalities across linear projections, aiming for near-isometric entropy behavior.

Known upper and lower bounds exist for specific slope sets; the conjecture posits asymptotic optimality near 1.

References

The arithmetic Kakeya conjecture asserts that $C_{\ref{arith}({r_1,\dots,r_k}; r_\infty)$ can be made arbitrarily close to $1$.

— Mathematical exploration and discovery at scale (2511.02864 - Georgiev et al., 3 Nov 2025) in Subsection “The arithmetic Kakeya conjecture” (Section 4.14)