In this paper, we construct a total Boolean function with $\widetilde{O}(\log n)$ randomized communication protocol, while any monochromatic rectangle has density at most $O(2^{-\mathrm{poly}(n)})$. As a corollary, it gives the first total function separation for $\mathrm{BPP}\not\subseteq\mathrm{P}^{\mathrm{NP}}$ in the communication world. Inspired by Gavinsky’s recent work (arXiv:2608.18784), our construction combines the ... more >>>

For every fixed $k\ge3$, we construct an explicit total Boolean function in the $k$-player number-on-forehead model with public-coin randomized communication complexity $O_k(1)$ and nondeterministic communication complexity $\Omega_k(n)$, where $n$ is the number of bits on each forehead. This extends the explicit three-player separations of Kelley, Lovett, and Meka (STOC 2024) ... more >>>

Understanding the power and limitations of classical and quantum information, and how they differ, is an important endeavor. On the classical side, property testing of distributions is a fundamental task: a tester, given samples of a distribution over a typically large domain such as $\{0,1\}^n$, is asked to verify properties ... more >>>

We prove that for every decision tree, the absolute values of the Fourier coefficients of given order $\ell\geq1$ sum to at most $c^{\ell}\sqrt{{d\choose\ell}(1+\log n)^{\ell-1}},$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant. This bound is essentially tight and settles a ... more >>>

The threshold degree of a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is the minimum degree of a real polynomial $p$ that represents $f$ in sign: $\mathrm{sgn}\; p(x)=(-1)^{f(x)}.$ A related notion is sign-rank, defined for a Boolean matrix $F=[F_{ij}]$ as the minimum rank of a real matrix $M$ with $\mathrm{sgn}\; M_{ij}=(-1)^{F_{ij}}$. Determining the maximum ... more >>>

Interactive coding, pioneered by Schulman (FOCS 1992, STOC 1993), is concerned with making communication protocols resilient to adversarial noise. The canonical model allows the adversary to alter a small constant fraction of symbols, chosen at the adversary's discretion, as they pass through the communication channel. Braverman, Gelles, Mao, and Ostrovsky ... more >>>