期刊
MONATSHEFTE FUR MATHEMATIK
卷 189, 期 1, 页码 137-156出版社
SPRINGER WIEN
DOI: 10.1007/s00605-018-1199-2
关键词
Poissonian pair correlations; Additive energy; Diophantine approximation; Metric number theory
类别
资金
- Austrian Science Fund (FWF) [Y-901]
- Austrian Science Fund (FWF) Projects [W1230]
Let (an) n be a strictly increasing sequence of positive integers. Recentworks uncovered a close connection between the additive energy E (AN) of the cut-offs AN = {an : n = N}, and (an) n possessing metric Poissonian pair correlations which is a metric version of a uniform distribution property of second order. Firstly, the present article makes progress on a conjecture of Aichinger, Aistleitner, and Larcher; by sharpening a theorem of Bourgain which states that the set of a. [ 0, 1] satisfying that (aan ) n with E (AN) = N3 does not have Poissonian pair correlations has positive Lebesgue measure. Secondly, we construct sequences with high additive energy which do not have metric Poissonian pair correlations, in a strong sense, and provide Hausdorff dimension estimates.
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