Parallel Repetition for 3-Player XOR Games

In a 3-XOR game G, the verifier samples a challenge (x, y, z) where is a probability distribution over, and a map t 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 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.