Multiplicative Number Theory and Irregularities of Distribution

9800653贝克该奖项为调查以下六个问题提供资金。(I)具有几乎素变量的拉格朗日四平方定理；PI旨在通过使用具有更少素数因子的数来加强Brudern和Fuvry的结果。(Ii)Krousterman和与尖点形式的傅里叶系数；PI应重新研究J.M.Deshouillers和H.Iwaniec的这一经典著作，以期加强Krousterman和的和的适用界限。(Iii)Bartan-Davenport-Halberstam型和的下界。已知算术级数中素数的数量与期望值之间的差异在某些范围内至少是期望值的‘方差’。PI的目标是扩大这些范围。(Iv)短区间几乎素数：许多学者研究过的筛分问题，其中最多有两个素数因子的数必须证明在短区间内存在。(V)一组点相对于凸体的偏差。这里，N点集的预期成员数和实际成员数之间的差值取自具有光滑边界的凸集的扩张平移；Drmota有尖锐的下界，PI旨在削弱他的光滑条件。(Vi)点集相对于球的差异：取上述一组球的预期点数和实际点数之间的差值。PI试图强化霍尔特版本的鄂尔多斯-图兰定理，该定理将出现的最大差异与指数和的加权平均联系在一起。(其中一些问题将与格林·哈曼共同解决。)这项研究属于数论的一般数学领域。数论起源于对整体数的研究，主要研究素数、完美平方和等问题。它是最古老的数学分支之一，在17世纪费马的手中享受着创造力的激增，并从那时起不断成长和发展。在过去的半个世纪内，它已适用于数据传输和处理以及通信系统中的重要问题。

9800653 Baker This award provides funds for an investigation into the following six problems. (i) Lagrange's four squares theorem with almost prime variables; the PI aims to sharpen the results of Brudern and Fouvry by using numbers with far fewer prime factors. (ii) Kloosterman sums and Fourier coefficients of cusp forms; the PI shall reexamine this classic work of J. M. Deshouillers and H. Iwaniec with a view to sharpening the applicable bounds on sums of Kloosterman sums. (iii) Lower bounds for sums of Bartan-Davenport-Halberstam type. A 'variance' of differences between the number of primes in arithmetic progressions and the expected value is known to be at least of the expected order of magnitude in certain ranges. The PI aims to extend these ranges. (iv) Almost primes in short intervals: a sieve problem on which many scholars have worked, where a number with at most two prime factors must be shown to exist in a short interval. (v) Discrepancy of a set of points with respect to a convex body. Here the difference is taken between the 'expected' number and the actual number of members of an N-point set in a dilated translation of a convex set with smooth boundary; Drmota has sharp lower bounds and the PI aims to weaken his smoothness conditions. (vi) Discrepancy of point sets with respect to balls: the difference is taken between expected number and actual number of points of a set as above in a ball. The PI seeks to sharpen Holt's version of the Erdos-Turan theorem, which relates the largest difference that occurs to a weighted average of exponential sums. (Work on some of these problems will be joint with Glyn Harman.) This research falls into the general mathematical field of number theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with prime numbers, sums of perfect squares, and so on. It is among the oldest branches of mathematics, enjoying a surge of creativity at the hands of Fermat in the 17th century and having grown and developed continuously since then. Within the last half century it has become applicable to important problems in data transmission and processing, and communications systems.