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- W2753242305 abstract "Let $left(a_{n}right)_{n}$ be a strictly increasing sequence of positive integers, denote by $A_{N}=left{ a_{n}:,nleq Nright} $ its truncations, and let $alphainleft[0,1right]$. We prove that if the additive energy $Eleft(A_{N}right)$ of $A_{N}$ is in $Omegaleft(N^{3}right)$, then the sequence $left(leftlangle alpha a_{n}rightrangle right)_{n}$ of fractional parts of $alpha a_{n}$ does not have Poissonian pair correlations (PPC) for almost every $alpha$ in the sense of Lebesgue measure. Conversely, it is known that $Eleft(A_{N}right)=mathcal{O}left(N^{3-varepsilon}right)$, for some fixed $varepsilon>0$, implies that $left(leftlangle alpha a_{n}rightrangle right)_{n}$ has PPC for almost every $alpha$. This note makes a contribution to investigating the energy threshold for $Eleft(A_{N}right)$ to imply this metric distribution property. We establish, in particular, that there exist sequences $left(a_{n}right)_{n}$ with [ Eleft(A_{N}right)=Thetaleft(frac{N^{3}}{logleft(Nright)logleft(log Nright)}right) ] such that the set of $alpha$ for which $left(alpha a_{n}right)_{n}$ does not have PPC is of full Lebesgue measure. Moreover, we show that for any fixed $varepsilon>0$ there are sequences $left(a_{n}right)_{n}$ with $Eleft(A_{N}right)=Thetaleft(frac{N^{3}}{logleft(Nright)left(loglog Nright)^{1+varepsilon}}right)$ satisfying that the set of $alpha$ for which the sequence $left(bigllanglealpha a_{n}bigrrangleright)_{n}$ does not have PPC is of full Hausdorff dimension." @default.
- W2753242305 created "2017-09-15" @default.
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- W2753242305 date "2017-08-29" @default.
- W2753242305 modified "2023-09-25" @default.
- W2753242305 title "On Exceptional Sets in the Metric Poissonian Pair Correlations problem" @default.
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