Prof. Michael Elkin

The problem of constructing dense subsets S of {1, 2, . . . , n} that contain no three-term arithmetic progression was introduced by Erdocombining double acute accents and Turán in 1936. They have presented a construction with |S| = Ω(n ^{log3 2}) elements. Their construction was improved by Salem and Spencer, and further improved by Behrend in 1946. The lower bound of Behrend is (Chemical Equation Presented) Since then the problem became one of the most central, most fundamental, and most intensively studied problems in additive number theory. Nevertheless, no improvement of the lower bound of Behrend has been reported since 1946. In this paper we present a construction that improves the result of Behrend by a factor of Θ(√log n), and shows that (Chemical Equation Presented) In particular, our result implies that the construction of Behrend is not optimal. Our construction and proof are elementary and self-contained. Also, the construction can be implemented by an efficient algorithm. Behrend's construction has numerous applications in Theoretical Computer Science. In particular, it is used for fast matrix multiplication, for property testing, and in the area of communication complexity. Plugging in our construction instead of Behrend's construction in the matrix multiplication algorithm of Coppersmith and Winograd improves the state-of-the-art upper bound on the complexity of the matrix multiplication by a factor of log^υ n, for some fixed constant υ > 0. We also present an application of our technique in Computational Geometry.