Original Paper

Geometric & Functional Analysis GAFA

, Volume 16, Issue 2, pp 327-366

First online:

Exponential sum estimates over subgroups and almost subgroups of \( \mathbb{Z}_{Q}^{*} \), where Q is composite with few prime factors

  • J. BourgainAffiliated withInstitute of Advanced Study Email author 
  • , M. -C. ChangAffiliated withDepartment of Mathematics, University of California at Riverside

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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  =  p r (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].

Keywords and phrases.

Sum and product set exponential sum

2000 Mathematics Subject Classification.

11L07 11L05