Abstract.
In this paper we extend the exponential sum results from [BK] and [BGK] 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 qδ, for any given δ > 0. The method consists in first establishing a ‘sumproduct theorem’ for general subsets A of \(\mathbb{Z}^{q} \) . If q is prime, the statement, proven in [BKT], expresses simply that either the sum-set 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 [BGK] 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 get nontrivial bounds for the ‘Heilbronn-type’ exponential sums when q = pr (p prime) for all r. Only the case r = 2 has 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 [KS].
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Submitted: October 2004 Revision: June 2005 Accepted: August 2005
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Bourgain, J., Chang, M.C. Exponential sum estimates over subgroups and almost subgroups of \( \mathbb{Z}_{Q}^{*} \), where Q is composite with few prime factors. GAFA, Geom. funct. anal. 16, 327–366 (2006). https://doi.org/10.1007/s00039-006-0558-7
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DOI: https://doi.org/10.1007/s00039-006-0558-7