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

- First Online:

DOI: 10.1007/s00039-006-0558-7

- Cite this article as:
- Bourgain, J. & Chang, M.-. GAFA, Geom. funct. anal. (2006) 16: 327. doi:10.1007/s00039-006-0558-7

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## 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 *q*^{δ}, 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].