Georgia Institute of Technology College of Sciences

  We are interested in arithmetic progressions in positive measure subsets of [0,1]^d. After a counterexample by Bourgain, it seemed as if nothing could be said about the longest interval formed by sizes of their gaps. However, Cook, Magyar, and Pramanik gave a positive result for 3-term progressions if their gaps are measured in the l^p-norm for p other than 1, 2, and infinity, and the dimension d is large enough. We establish an appropriate generalization of their result to longer progressions. The main difficulty lies in handling a class of multilinear singular integrals associated with arithmetic progressions that includes the well-known multilinear Hilbert transforms, bounds for which still constitute an open problem. As a substitute, we use the previous work with Durcik and Thiele on power-type cancellation of those transforms, which was, in turn, motivated by a desire to quantify the results of Tao and Zorin-Kranich. This is joint work with Polona Durcik (Caltech).