The distribution of spacings between the fractional parts of \(n^2\alpha\)
Let \(\alpha\) be an irrational number. The problem of the distribution of the local spacings between the members of the sequence \(n^{2}\alpha\) mod \(1, 1 \leq n \leq N\) arises in the study of the local spacing distributions between the eigenvalues of special Hamiltonians. Denote by \(0 \leq \beta_{1} \leq \beta_{2}\leq \dots \leq \beta_{N} < 1, \beta_{N+j}= \beta_{j}\) an ordering of the numbers \(n^{2}\alpha\) mod \(1.\) The \(k\)th consecutive spacing measure is defined by \[ \mu_{k}(N,\alpha): = \frac{1}{N}\sum^{N}_{j=1}\delta_{N(\beta_{j+k}-\beta_{j})}, \] where \(\delta_{x}\) is a unit delta mass at \(x.\) The problem is to understand the behavior of these measures as \(N \to \infty\) and their dependence on the Diophantine approximations to \(\alpha.\) The authors in particular investigate the Poissonian property of the corresponding correlations. Poissonian in this sense implies that as far as local spacings go, the numbers behave randomly. A typical result is as follows: Let \(\alpha \in \mathbb{R}.\) Suppose there are infinitely many rationals \(b_{j}/q_{j},\) with \(q_{j}\) prime, satisfying \[ \left|\alpha - \frac{b_{j}}{q_{j}}\right|< \frac{1}{q^{3}_{j}}. \] Then there is a subsequence \(N_{j}\to \infty\) with \(\frac{\log N_{j}}{\log q_{j}}\to 1\) for which \(n^{2}\alpha\) mod \(1\) is Poissonian along this subsequence. Furthermore the authors formulate the following Conjecture: If \(\alpha\) is of approximation type \(2+ \varepsilon\) for every \(\varepsilon > 0\) and the convergents \(\frac{a}{q}\) to \(\alpha\) satisfy \(\lim_{q\to\infty} \frac{\log \tilde{q}}{\log q}= 1,\) where \(\tilde{q}\) is the square free part of \(q,\) then \(n^{2}\alpha\) mod \(1\) is Poissonian. The conjecture is true for almost all \(\alpha\) (in the sense of Lebesgue measure) and for real algebraic \(\alpha\) (provided that the \(abc\)-conjecture holds).
scientific article; zbMATH DE number 1655740