Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems
Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems
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度量对相关问题中例外集的加性能量和豪斯多夫维数
DOI：
10.1007/s11856-017-1597-5
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发表时间：
2016
影响因子：
1
通讯作者：
Mark Lewko
中科院分区：
文献类型：
--
作者：
C. Aistleitner;G. Larcher;Mark Lewko
对于整数序列{a(x)x}≥1，我们证明了如果序列截断的加性能量比平凡估计有更大的节省功率，则{< αa(x) >} x≥1的分数部分的对相关分布对于几乎所有α都是渐近泊松的。此外，我们给出了异常集的Hausdorff维数作为序列密度和能量估计中节省的功率的函数的估计。这些结果的一个结果是，使{< αxd >}不具有泊松对相关的α集合的Hausdorff维数至多为$$\frac{{d + 2}}{{d + 3}} < 1$$ d+2d+3<1。这加强了Rudnick和Sarnak的结论，即异常集的勒贝格测度为零。另一方面，经典的例子表明，例外集的豪斯多夫维数至少为$$\frac{2}{{d + 1}}$$ 2d+1。Jean Bourgain在本文第一版完成后添加了附录。本文在附录中解决了本文提出的两个问题。
For a sequence of integers {a(x)}x≥1 we show that the distribution of the pair correlations of the fractional parts of {〈αa(x)〉}x≥1 is asymptotically Poissonian for almost all α if the additive energy of truncations of the sequence has a power savings improvement over the trivial estimate. Furthermore, we give an estimate for the Hausdorff dimension of the exceptional set as a function of the density of the sequence and the power savings in the energy estimate. A consequence of these results is that the Hausdorff dimension of the set of α such that {〈αxd〉} fails to have Poissonian pair correlation is at most $$\frac{{d + 2}}{{d + 3}} < 1$$d+2d+3<1. This strengthens a result of Rudnick and Sarnak which states that the exceptional set has zero Lebesgue measure. On the other hand, classical examples imply that the exceptional set has Hausdorff dimension at least $$\frac{2}{{d + 1}}$$2d+1.An appendix by Jean Bourgain was added after the first version of this paper was written. In this appendix two problems raised in the paper are solved.