Sum-product theorem and exponential sum estimates in residue classes with modulus involving few prime factors

Abstract

In this Note, we extend the results of Bourgain, Konyagin and Glibichuk to certain composite moduli q involving few ‘large’ primes. First a ‘sum-product’ theorem for subsets A of Z q is obtained, ensuring that | A + A | + | A . A | > c | A | 1 + ɛ provided | A | < q 1 − δ and A does not have a ‘large’ intersection with a translate of a subring. Next, exponential sum estimates are established. In particular nontrivial bounds are obtained for the exponential sums associated to a multiplicative subgroup H < Z q * , with applications to Heilbronn-type sums. To cite this article: J. Bourgain, M.-C. Chang, C. R. Acad. Sci. Paris, Ser. I 339 (2004).