Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order. (English. Abridged French version) Zbl 1041.11056

Bourgain, Jean; Konyagin, S. V.

C. R., Math., Acad. Sci. Paris 337, No. 2, 75-80 (2003).

Let \(A\) be a subset of \(F={\mathbb Z}/p{\mathbb Z}\) and write \(A+A=\{a+b:a,b\;\text{ in}\;A\}, A.A=\{ab:a,b\;\text{ in}\;A\}\) and \(| A| \) for the cardinality of \(A\). The first estimate is \(\max(| A+A| ,| A.A| )\geq c_1| A| ^{1+c_2}\) if \(| A| <p^{1/2}\), for some constants \(c_1, c_2>0\).
Let \(e(u)=e^{2\pi iu}\), \(G\) be a subgroup of \(F^*\) and \(S(G)= \max_{\xi \in F^*}\left| \sum_{x\in G}e\left({x\xi\over p} \right)\right| .\)
The second estimate is \(S(G)\leq | G| P^{-\gamma}\) with \(\gamma=\exp(-C_1/\delta^{C_2})\) if \(\delta>0\) and \(| G| \geq p^\delta\), for some constants \(C_1, C_2>0\). The paper includes statements of several interesting auxiliary results and sketches of some of the proofs.

Reviewer: John H. Loxton (North Ryde)

MSC:

11L07	Estimates on exponential sums
11T23	Exponential sums

Keywords:

sum sets; product sets; exponential sums

References:

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