Parallel Repetition for 3-Player XOR Games

In a 3-XOR game G, the verifier samples a challenge (x, y, z) ∼ µ where µ is a probability distribution over Σ × Γ × Φ, and a map t : Σ × Γ × Φ → A for a finite Abelian group A defining a constraint. The verifier sends the questions x, y and z to the players Alice, Bob and Charlie respectively, receives answers f (x), g(y) and h(z) that are elements in A and accepts if f (x) + g(y) + h(z) = t(x, y, z). The value, val(G), of the game is defined to be the maximum probability the verifier accepts over all players' strategies. We show that if G is a 3-XOR game with value strictly less than 1, whose underlying distribution over questions µ does not admit Abelian embeddings into (Z, +), then the value of the n-fold repetition of G is exponentially decaying. That is, there exists c = c(G) > 0 such that val(G ⊗n ) ≤ 2 -cn . This extends a previous result of [Braverman-Khot-Minzer, FOCS 2023] showing exponential decay for the GHZ game. Our proof combines tools from additive combinatorics and tools from discrete Fourier analysis.