Entry and return times for semi-flows

Haydn, Lacroix and Vaienti [Ann. Probab. 33 (2005)] proved that, for a given ergodic map, the entry time distribution converges in the small target limit, if and only if the corresponding return time distribution converges. The present note explains how entry and return times can be interpreted in terms of stationary point processes and their Palm distribution. This permits a generalization of the results by Haydn et al. to non-ergodic maps and continuous-time dynamical systems.Comment: 14 pages; to appear in Nonlinearit

Pair correlation and equidistribution on manifolds

This study is motivated by a series of recent papers that show that, if a given deterministic sequence in the unit interval has a Poisson pair correlation function, then the sequence is uniformly distributed. Analogous results have been proved for point sequences on higher-dimensional tori. The purpose of this paper is to describe a simple statistical argument that explains this observation and furthermore permits a generalisation to bounded Euclidean domains as well as compact Riemannian manifolds.Comment: 10 pages; to appear in Monatshefte fuer Mathemati

Pair correlation densities of inhomogeneous quadratic forms II

Denote by $\| \cdot \|$ the euclidean norm in \RR^k. We prove that the local pair correlation density of the sequence \| \vecm -\vecalf \|^k, \vecm\in\ZZ^k, is that of a Poisson process, under diophantine conditions on the fixed vector \vecalf\in\RR^k: in dimension two, vectors \vecalf of any diophantine type are admissible; in higher dimensions ($k>2$), Poisson statistics are only observed for diophantine vectors of type $\kappa<(k-1)/(k-2)$. Our findings support a conjecture of Berry and Tabor on the Poisson nature of spectral correlations in quantized integrable systems

Holomorphic almost modular forms

Holomorphic almost modular forms are holomorphic functions of the complex upper half plane which can be approximated arbitrarily well (in a suitable sense) by modular forms of congruence subgroups of large index in \SL(2,\ZZ). It is proved that such functions have a rotation-invariant limit distribution when the argument approaches the real axis. An example for a holomorphic almost modular form is the logarithm of \prod_{n=1}^\infty (1-\exp(2\pi\i n^2 z)). The paper is motivated by the author's studies [J. Marklof, Int. Math. Res. Not. {\bf 39} (2003) 2131-2151] on the connection between almost modular functions and the distribution of the sequence $n^2x$ modulo one

Pair correlation densities of inhomogeneous quadratic forms

Under explicit diophantine conditions on (\alpha,\beta)\in\RR^2, we prove that the local two-point correlations of the sequence given by the values (m-\alpha)^2+\break (n-\beta)^2, with (m,n)\in\ZZ^2, are those of a Poisson process. This partly confirms a conjecture of Berry and Tabor [2] on spectral statistics of quantized integrable systems, and also establishes a particular case of the quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms of signature (2,2). The proof uses theta sums and Ratner's classification of measures invariant under unipotent flows.Comment: 53 pages published versio

Quantum Leaks

We show that eigenfunctions of the Laplacian on certain non-compact domains with finite area may localize at infinity--provided there is no extreme level clustering--and thus rule out quantum unique ergodicity for such systems. The construction is elementary and based on `bouncing ball' quasimodes whose discrepancy is proved to be significantly smaller than the mean level spacing

Gaps between logs

We calculate the limiting gap distribution for the fractional parts of log n, where n runs through all positive integers. By rescaling the sequence, the proof quickly reduces to an argument used by Barra and Gaspard in the context of level spacing statistics for quantum graphs. The key ingredient is Weyl equidistribution of irrational translations on multi-dimensional tori. Our results extend to logarithms with arbitrary base; we deduce explicit formulas when the base is transcendental or the r:th root of an integer. If the base is close to one, the gap distribution is close to the exponential distribution.Comment: 14 page