Understanding Parallel Repetition Requires Understanding Foams

Motivated by the study of parallel repetition and also by the unique games conjecture, we investigate the value of the "odd cycle games" under parallel repetition. Using tools from discrete harmonic analysis, we show that after d rounds on the cycle of length m, the value of the game is at most 1-(1/m)ldrOmega macr(radicd) (for dlesm
²
, say). This beats the natural barrier of 1-Theta(1/m)
²
ldrd for Raz-style proofs and also the SDP bound of Feige-Lovasz; however, it just barely fails to have implications for unique games. On the other hand, we also show that improving our bound would require proving nontrivial lower bounds on the surface area of high-dimensional foams. Specifically, one would need to answer: what is the least surface area of a cell that tiles R
^d
by the lattice Z
^d
?