EXPONENTIAL SUM ESTIMATES OVER SUBGROUPS AND ALMOST SUBGROUPS OF Zq, WHERE q IS COMPOSITE WITH FEW PRIME FACTORS
EXPONENTIAL SUM ESTIMATES OVER SUBGROUPS AND ALMOST SUBGROUPS OF Zq, WHERE q IS COMPOSITE WITH FEW PRIME FACTORS
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Zq 的子群和几乎子群的指数和估计，其中 q 是具有很少素因子的合数
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发表时间：
2006
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通讯作者：
Mei
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作者：
J. Bourgain;Mei
本文将[B-K]和[B-G-K]关于素模的指数和的结果推广到含有有限素数因子的复合模q。特别地，对于任意给定的δ > 0，我们得到了与大小为q的乘法子群H相关的指数和的非平凡界。该方法包括首先为z的一般子集a建立一个“和积定理”。如果q是素数，则在[B-K-T]中证明的陈述简单地表示，除非| a |靠近q，否则sumset a + a或product-set A.A显著大于a。对于复合q，非平凡子的存在需要更复杂的二分法，这在这里建立。有了这个和积定理，[B-G-K]中的方法就可以适用于复合模的当前情况。他们主要依赖于谐波分析和图理论结果，如高尔斯的定量版巴洛格-塞梅雷迪定理。作为推论，当所有r的q = p （p ‘）时，我们确实得到了’ heilbronn型'指数和的非平凡界限。只有r = 2的情况在Heath-Brown和Heath-Brown和Konyagin（使用Stepanov的方法）的早期作品中得到了处理。我们也得到了涉及指数函数的（可能不完整的）和的指数和估计，例如在[Konyagin-Shparlinski]中所考虑的。
In this paper we extend the exponential sum results from [B-K] and [B-G-K] 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, for any given δ > 0. The method consists in first establishing a ‘sum-product theorem’ for general subsets A of Z. If q is prime, the statement, proven in [B-K-T], expresses simply that, either the sumset 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 [B-G-K] 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 do get nontrivial bounds for the ‘Heilbronn-type’ exponential sums when q = p (p prime) for all r. Only the case r = 2 had 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 [Konyagin-Shparlinski]