New Bounds on Cap Sets

The problem of the maximal size of cap sets is a characteristic 3 model for the problem of finding arithmetic progressions of length 3 in rather dense sets of integers. Meshulam [M95] , through a direct use of ideas of Roth, was able to show that there is a constant C so that any cap set A has density at most C N . Our result may be viewed as a very modest improvement over Meshulam’s result. Sanders [S11] recently showed that any subset of the integers whose density in {1, . . . ,M} is at least C(log logM) 5 logM must contain arithmetic progressions of length 3. This may be thought of as bringing the results known for arithmetic progressions almost to the level of Meshulam’s result. This has spurred further interest in improving Meshulam’s result in hopes that it might suggest a way of improving the results on arithmetic progressions. A rather concrete, though perhaps still out of reach, goal in this direction is a conjecture of Erdös and Turan: