The Arithmetic Kakeya Conjecture

The problem

For each slope $r\in \mathbb{R}\cup \{\infty \}$ define the projection ${\pi }_r:{\mathbb{R}}^2\to \mathbb{R}$ by ${\pi }_r(a,b)=a+rb$ for $r\ne \infty$ and ${\pi }_{\infty }(a,b)=b$. Given a set $r_1,\dots ,r_k,r_{\infty }$ of distinct slopes, we let $C(\{r_1,\dots ,r_k\};r_{\infty })$ be the smallest constant for which the following is true: if $X,Y$ are discrete random variables (not necessarily independent) taking values in a finite set of reals, then $$\mathbf{H}({\pi }_{r_{\infty }}(X,Y))\le C(\{r_1,\dots ,r_k\};r_{\infty })\underset{i=1,\dots ,k}{\max }\mathbf{H}({\pi }_{r_i}(X,Y)),$$ where $\mathbf{H}(X)=-{\sum }_{x}P(X=x)\log (P(X=x))$ is the entropy of a random variable and $x$ ranges over the values taken by $X$. The arithmetic Kakeya conjecture asserts that $C(\{r_1,\dots ,r_k\};r_{\infty })$ can be made arbitrarily close to $1$.

AlphaEvolve found a construction better than the previous record (at publication).

What counts as progress

• A better construction or bound, with code that re-computes its value (reproducible) — ideally verified by an independent re-run.
• A proof that a known construction is optimal, or a better bound on the other side.
• Literature: earlier or newer records (literature claims).

Source and licence

Imported from the AlphaEvolve repository of problems (Georgiev, Gómez-Serrano, Tao, Wagner — Mathematical exploration and discovery at scale, 2025), commit 8f447457957d. Text under CC BY 4.0, code under Apache 2.0; reformatted for this page.