We study the power of randomness in the Number-on-Forehead (NOF) model in communication complexity. We construct an explicit 3-player function $f:[N]^3 \to \{0,1\}$, such that: (i) there exist a randomized NOF protocol computing it that sends a constant number of bits; but (ii) any deterministic or nondeterministic NOF protocol computing it requires sending about $(\log N)^{1/3}$ many bits. This exponentially improves upon the previously best-known such separation. At the core of our proof is an extension of a recent result of the first and third authors on sets of integers without 3-term arithmetic progressions into a non-arithmetic setting.

翻译：我们研究通信复杂度中前额数（NOF）模型下随机性的能力。我们构造了一个显式的3方函数 $f:[N]^3 \to \{0,1\}$，满足： (i) 存在一个计算该函数的随机化NOF协议，仅需发送常数比特；但 (ii) 任何计算该函数的确定性或非确定性NOF协议需要发送约 $(\log N)^{1/3}$ 比特。这指数级地改进了先前已知的最佳分离结果。我们证明的核心是将第一和第三作者最近关于无三项等差数列整数集的结论推广到非算术场景中。