On exceptional sets in the metric Poissonian pair correlations problem
On exceptional sets in the metric Poissonian pair correlations problem
复制标题
关于度量泊松对相关问题中的异常集
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Niclas Technau
中科院分区：
文献类型：
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作者：
T. Lachmann;Niclas Technau
Let $$left( a_{n}
ight) _{n}$$ann be a strictly increasing sequence of positive integers. Recent works uncovered a close connection between the additive energy $$Eleft( A_{N}
ight) $$EAN of the cut-offs $$A_{N}=left{ a_{n},{:},,nle N
ight} $$AN=an:n≤N, and $$left( a_{n}
ight) _{n}$$ann 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 $$alpha in left[ 0,1
ight] $$α∈0,1 satisfying that $$left( leftlangle alpha a_{n}
ight
angle
ight) _{n}$$αann with $$Eleft( A_{N}
ight) =Omega left( N^{3}
ight) $$EAN=ΩN3 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.
Let $$left( a_{n}
ight) _{n}$$ann be a strictly increasing sequence of positive integers. Recent works uncovered a close connection between the additive energy $$Eleft( A_{N}
ight) $$EAN of the cut-offs $$A_{N}=left{ a_{n},{:},,nle N
ight} $$AN=an:n≤N, and $$left( a_{n}
ight) _{n}$$ann 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 $$alpha in left[ 0,1
ight] $$α∈0,1 satisfying that $$left( leftlangle alpha a_{n}
ight
angle
ight) _{n}$$αann with $$Eleft( A_{N}
ight) =Omega left( N^{3}
ight) $$EAN=ΩN3 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.