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on The Arithmetic Kakeya Conjecture (AlphaEvolve problem 30)

The statement

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 $1$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.