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：我们研究了通信复杂性中额上数字（NOF）模型里随机性的能力。我们构造了一个显式的三玩家函数 $f:[N]^3 \to \{0,1\}$，满足：(i) 存在一个计算该函数的随机化NOF协议，仅需发送常数比特数；但 (ii) 任何计算该函数的确定性或非确定性NOF协议都需要发送约 $(\log N)^{1/3}$ 比特。这比此前已知的最佳此类分离结果实现了指数级改进。我们证明的核心在于将第一和第三作者关于无三项等差数列整数集的最新成果推广至非算术设定。