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Full Poissonian local statistics of slowly growing sequences
- Part of
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- Journal:
- Compositio Mathematica / Volume 161 / Issue 1 / January 2025
- Published online by Cambridge University Press:
- 24 March 2025, pp. 148-180
- Print publication:
- January 2025
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- Article
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Fix
$\alpha >0$. Then by Fejér's theorem 
$(\alpha (\log n)^{A}\,\mathrm {mod}\,1)_{n\geq 1}$ is uniformly distributed if and only if 
$A>1$. We sharpen this by showing that all correlation functions, and hence the gap distribution, are Poissonian provided 
$A>1$. This is the first example of a deterministic sequence modulo 
$1$ whose gap distribution and all of whose correlations are proven to be Poissonian. The range of 
$A$ is optimal and complements a result of Marklof and Strömbergsson who found the limiting gap distribution of 
$(\log (n)\, \mathrm {mod}\,1)$, which is necessarily not Poissonian.
