The communication complexity of XOR games via summing operators

The discrepancy method is widely used to find lower bounds for communication complexity of XOR games. It is well known that these bounds can be far from optimal. In this context Disjointness is usually mentioned as a case where the method fails to give good bounds, because the increment of the value of the game is linear (rather than exponential) in the number of communicated bits. We show in this paper the existence of XOR games where the discrepancy method yields bounds as poor as one desires. Indeed, we show the existence of such games with any previously prescribed value. Specifically we prove the following: For any number of bits c and every 0 < δ < 1 and for every ǫ > 0, we show the existence of a XOR game such that its value, both without communication or with the use of c bits of communication, is contained in the interval (δ − ǫ, δ + ǫ). To prove this result we apply the theory of p-summing operators, a central topic in Banach space theory. We show in the paper other applications of this theory to the study of the communication complexity of XOR games.