TAILIEUCHUNG - Đề tài " An uncertainty principle for arithmetic sequences "

Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are “well-distributed” in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equidistribution, as have Fourier analysts when working with the “uncertainty principle”. In this article we find that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory, formulating a general principle, and giving several examples. . | Annals of Mathematics An uncertainty principle for arithmetic sequences By Andrew Granville and K. Soundararajan Annals of Mathematics 165 2007 593 635 An uncertainty principle for arithmetic sequences By Andrew Granville and K. SouNDARARAJAN Abstract Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are well-distributed in some appropriate sense. In various discrepancy problems combinatorics researchers have analyzed limitations to equidistribution as have Fourier analysts when working with the uncertainty principle . In this article we find that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory formulating a general principle and giving several examples. 1. Introduction In this paper we investigate the limitations to the equidistribution of interesting arithmetic sequences in arithmetic progressions and short intervals. Our discussions are motivated by a general result of K. F. Roth 15 on irregularities of distribution and a particular result of H. Maier 11 which imposes restrictions on the equidistribution of primes. If A is a subset of the integers in 1 x with A px then as Roth proved there exists N x and an arithmetic progression a mod q with q ựx such that I 1 - q 11 .4. nEA n N nEA n a mod q n N In other words keeping away from sets of density 0 or 1 there must be an arithmetic progression in which the number of elements of A is a little different from the average. Following work of A. Sarkozy and J. Beck J. Matousek and J. Spencer 12 showed that Roth s theorem is best possible in that there is a Le premier auteur est partiellement soutenu par une bourse du Conseil de recherches en sciences naturelles et en genie du Canada. The second author is partially supported by the National Science Foundation. 594 ANDREW GRANVILLE AND K. SOUNDARARAJAN set A containing x 2 integers up to x for which n E A n N n a mod q n E A n N q x1 4 for all q and a with N x. Roth

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