AlphaEvolve problems·alphaevolve:30

The Arithmetic Kakeya Conjecture

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 $1$.

Source

Something wrong or missing here? Request a correction or add a claim, with its sources.