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).

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