Geometric & Functional Analysis GAFA

, Volume 16, Issue 2, pp 327–366

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

Original Paper


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].

Keywords and phrases.

Sum and product set exponential sum 

2000 Mathematics Subject Classification.

11L07 11L05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Institute of Advanced StudyPrincetonUSA
  2. 2.Department of MathematicsUniversity of California at RiversideUSA

Personalised recommendations