Some negative results related to Poissonian pair correlation problems
Gerhard Larcher; Wolfgang Stockinger · 2018 · arXiv
WASTE classifies this as Negative / Null Result Report · AI classification, approximate
The study found no significant effect — useful as a negative control or null benchmark for your own design.
Abstract (excerpt)
We say that a sequence $(x_n)_{n \in \mathbb{N}}$ in $[0,1)$ has Poissonian pair correlations if \begin{equation*} \lim_{N \to \infty} \frac{1}{N} \# \left \lbrace 1 \leq l \neq m \leq N: \| x_l - x_m \| \leq \frac{s}{N} \right \rbrace = 2s \end{equation*} for every $s \geq 0$. The aim of this article is twofold. First, we will establish a gap theorem which allows to deduce that a sequence $(x_n)_{n \in \mathbb{N}}$ of real numbers in $[0,1)$ having a certain weak gap structure, cannot have Poissonian pair correlations. This result covers a broad class of sequences, e.g., Kronecker sequences,
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Metadata source: arXiv