Marcel K. Goh

Marcel K. Goh, Hamed Hatami

It is conjectured that every idempotent Schur multiplier can be written as a finite sum of contractive idempotents. This conjecture is equivalent to the statement that any boolean matrix $A$ with factorization norm $\lVert A\rVert_{γ_2}$ at most $γ$ can be expressed as a signed sum $$A = \sum_{i=1}^L \pm B_i,$$ where, up to permutation of rows and columns, each $B_i$ is a blow-up of an identity matrix, and $L$ depends only on $γ$. In this note we show that if $A$ is an $n\times n$ boolean matrix with $\lVert A\rVert_{γ_2} \le γ$, then it admits such an expression with $L = 2^{O(γ^9) + \log^*\! n}$, where $\log^*$ is the iterated logarithm function. As an application, any sequence of matrices with bounded factorization norm belongs to the complexity class $\mathrm{P}^\mathrm{EQ}$ of communication problems with polylogarithmic equality-oracle complexity.

Lampros Gavalakis, Marcel K. Goh, Ioannis Kontoyiannis

Various lower bounds are established for the entropy of sums, products and their combinations. First, we derive a prime-field analogue of a version of the entropy power inequality established by Tao over torsion-free groups. Next, we prove an entropy sum-product statement: For independent and identically distributed random variables $X,X'$, the maximum of ${\bf H}(X+X')$ and ${\bf H}(XX')$ is bounded below by a linear combination of the entropy and the min-entropy (Rényi entropy of order~$\infty$) of $X$. This result, obtained by bounding entropies of the form ${\bf H}\bigl( X(Y+Z)\bigr)$ from above and below, is valid over arbitrary fields $F$. Over $F={\bf R}$, a slightly stronger inequality is derived. Finally, a weak version of a purely Shannon-entropic sum-product result is developed: If the entropic additive doubling of a random variable $X$ over an arbitrary field is $O(1)$, then its multiplicative doubling is at least proportional to ${\bf H}(X)$.

Marcel K. Goh, Hamed Hatami

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.