EXPONENTIAL SUM ESTIMATES OVER SUBGROUPS AND ALMOST SUBGROUPS OF Zq, WHERE q IS COMPOSITE WITH FEW PRIME FACTORS

In this paper we extend the exponential sum results from [B-K] and [B-G-K] for prime moduli to composite moduli q involving a bounded number of prime factors. In particular, we obtain nontrivial bounds on the exponential sums associated to multiplicative subgroups H of size q, for any given δ > 0. The method consists in first establishing a ‘sum-product theorem’ for general subsets A of Z. If q is prime, the statement, proven in [B-K-T], expresses simply that, either the sumset A + A or the product-set A.A is significantly larger than A, unless |A| is near q. For composite q, the presence of nontrivial subrings requires a more complicated dichotomy, which is established here. With this sum-product theorem at hand,the methods from [B-G-K] may then be adapted to the present context with composite moduli. They rely essentially on harmonic analysis and graph-theoretical results such as Gowers’ quantitative version of the Balog-Szemeredi theorem. As a corollary,we do get nontrivial bounds for the ‘Heilbronn-type’ exponential sums when q = p (p prime) for all r. Only the case r = 2 had been treated earlier in works of Heath-Brown and Heath-Brown and Konyagin (using Stepanov’s method). We also get exponential sum estimates for (possibly incomplete) sums involving exponential functions, as considered for instance in [Konyagin-Shparlinski]