We construct a point-line incidence problem over the reals whose randomized communication complexity is constant, but whose deterministic communication complexity is linear even when the players have access to an equality oracle. This is the strongest possible separation between these two measures, and it improves on an earlier $O(1)$-versus-$Ω(\sqrt{n})$ separation of Göös, Harms, and Riazanov. Because point-line incidence problems have constant sign rank, our construction also bears on a question of Harms and Zamaraev, who asked whether constant sign rank together with constant randomized communication complexity forces constant equality-oracle complexity. This was already refuted by Göös, Harms, Imbach, and Sokolov with a logarithmic lower bound; our example improves the separation to linear, which is optimal. The proof draws on a construction in the recent disproof of the sum-product conjecture over the reals by Bloom, Sawin, Schildkraut, and Zhelezov, using totally real number fields of large degree and small discriminant. 12 pages, including references
We study property testing in the subcube conditional model introduced by Bhattacharyya and Chakraborty (2017). We obtain the first equivalence test for $n$-dimensional distributions that is quasi-linear in $n$, improving the previously known $\tilde{O}(n^2/\varepsilon^2)$ query complexity bound to $\tilde{O}(n/\varepsilon^2)$. We extend this result to general finite alphabets with logarithmic cost in the alphabet size. By exploiting the specific structure of the queries that we use (which are more restrictive than general subcube queries), we obtain a cubic improvement over the best known test for distributions over $\{1,\ldots,N\}$ under the interval querying model of Canonne, Ron and Servedio (2015), attaining a query complexity of $\tilde{O}((\log N)/\varepsilon^2)$, which for fixed $\varepsilon$ almost matches the known lower bound of $Ω((\log N)/\log\log N)$. We also derive a product test for $n$-dimensional distributions with $\tilde{O}(n / \varepsilon^2)$ queries, and provide an $Ω(\sqrt{n} / \varepsilon^2)$ lower bound for this property.
