Gap-Majority Lemmas in Communication Complexity

We prove an information-theoretically optimal \emph{gap-majority lemma} in the two-player randomized communication model. For a base function $f: \mathcal{X} \to \{\pm 1\}$, its $n$-fold \emph{gap-majority composition}, denoted $\mathsf{GapMAJ} \circ f^n$, takes $n$ inputs $(X_1, \ldots, X_n)$ and distinguishes whether $f^{+n}(X_1,\ldots,X_n) := f(X_1) + \ldots + f(X_n)$ is at least $0.01\sqrt{n}$ or at most $-0.01\sqrt{n}$. We show that if computing $f$ with success probability $0.501$ requires $I$ bits of information, then computing $\mathsf{GapMAJ} \circ f^n$ with success probability $0.99$ requires $n \cdot (I - O(1))$ bits of information. This result is asymptotically optimal in two aspects: it achieves the correct linear scaling of information cost and the correct constant-constant tradeoff between error rates. This makes $\mathsf{GapMAJ}$, to our knowledge, only the third explicit outer gadget that admits a strong composition theorem in the two-player communication setting, following the identity and XOR gadgets. From an application side, our gap-majority lemma can be viewed as a generic amplification tool that lifts the hardness of deciding $f$ into the hardness of approximating $f^{+n}$. Using this framework, we give a new proof to the communication lower bound of Gap-Hamming and derive a tight streaming lower bound of triangle counting, demonstrating the versatility of the gap-majority lemma.