Statistics of gaps in the sequence \(\{\sqrt{n}\}\)
Let \(0 < \alpha < 1\), \(N \in \mathbb{N}\) and define the set NEWLINE\[NEWLINE \Omega_\alpha(N) = \big\{ \{n^\alpha\}; \; 1 \leq n \leq N \big\} \subset [0,1] NEWLINE\]NEWLINE where \(\{b\}\) denotes the fractional part of \(b\). It was proposed by \textit{M. D.\ Boshernitzan} [J. Anal. Math. 62, 225--240 (1994; Zbl 0804.11046)] to study statistical properties of the normalized distance distribution between neighbouring points in \(\Omega_\alpha(N)\) as \(N \to \infty\). \textit{N. D.\ Elkies} and \textit{C. T.\ McMullen} [Duke Math. J. 123, No. 1, 95--139 (2004; Zbl 1063.11020)] solved the case \(\alpha= \frac{1}{2}\).NEWLINENEWLINEThe author shows a different approach to find the (existence of the) limiting distribution for \(\alpha= \frac{1}{2}\), based on methods from [\textit{Y. G.\ Sinai} and \textit{C. Ulcigrai}, Ergodic Theory Dyn. Syst. 28, No. 2, 643--655 (2008; Zbl 1151.37010)].NEWLINENEWLINEFor the entire collection see [Zbl 1237.37004].
scientific article; zbMATH DE number 6077503