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 $r \neq \infty$ 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

$${\mathbf H}(\pi_{r_\infty}(X,Y)) \leq C(\{r_1,\dots,r_k\}; r_\infty) \max_{i=1,\dots,k} {\mathbf H}(\pi_{r_i}(X,Y)),$$

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 $1$.

Statement quoted from problem 30 of the AlphaEvolve repository of problems (Google DeepMind, CC BY 4.0).

What AI did

AlphaEvolve found a construction that improved on the best bound known at the time; the problem itself remains open. Status in the repository: New result. See Section 6.15, in "Mathematical Exploration and Discovery at Scale" paper. Records may have moved since; the repository's notes below list the latest bounds it knows of.

How it was checked

The results are explicit constructions or programs; the repository's notebooks include the verification code used to score them.

Limitations

An improved bound is progress, not a solution: the exact answer to the problem is still unknown. The authors report that results were better when experts set up and guided the search.

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