The Arithmetic Kakeya Conjecture

AlphaEvolve found a construction that improved on the best bound known at the time; the problem itself remains open.

Problem
Problem 30 in the AlphaEvolve repository: The Arithmetic Kakeya Conjecture
Credit
AlphaEvolve (Google DeepMind) · LLM-guided evolutionary code search, with Georgiev, Gómez-Serrano, Tao and Wagner
Checked
Construction can be checked directlyPreprint (arXiv, 2025)
Timeline
Paper submitted 2025-11-03
Sources
Problem page (AlphaEvolve repository)Paper (arXiv:2511.02864)
PARTIALImproved bound · new construction

The problem

For each slope rinR∪{∞}r in \mathbb{R} \cup \{\infty\} define the projection πr:R2→R\pi_r : \mathbb{R}^2 \to \mathbb{R} by πr(a,b)=a+rb\pi_r(a,b) = a + rb for r≠∞r \neq \infty and π∞(a,b)=b\pi_\infty(a,b)=b. Given a set r1,…,rk,r∞r_1,\dots,r_k, r_\infty of distinct slopes, we let C({r1,…,rk};r∞)C(\{r_1,\dots,r_k\}; r_\infty) be the smallest constant for which the following is true: if X,YX,Y are discrete random variables (not necessarily independent) taking values in a finite set of reals, then

H(πr∞(X,Y))≤C({r1,…,rk};r∞)max⁡i=1,…,kH(πri(X,Y)),{\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 H(X)=−∑xP(X=x)log⁡(P(X=x)) {\mathbf H}(X) = -\sum_{x} P(X = x) \log(P(X=x)) is the entropy of a random variable and xx ranges over the values taken by XX. The arithmetic Kakeya conjecture asserts that C({r1,…,rk};r∞)C(\{r_1,\dots,r_k\}; r_\infty) can be made arbitrarily close to 11.

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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