The distribution of spacings between quadratic residues, II

We study the distribution of spacings between squares modulo q as the number of prime divisors of q tends to infinity. In an earlier paper Kurlberg and Rudnick proved that the spacing distribution for square free q is Poissonian, this paper extends the result to arbitrary q.Comment: Submitted for publication. 16 page

On the order of unimodular matrices modulo integers

Assuming the Generalized Riemann Hypothesis, we prove the following: If b is an integer greater than one, then the multiplicative order of b modulo N is larger than N^(1-\epsilon) for all N in a density one subset of the integers. If A is a hyperbolic unimodular matrix with integer coefficients, then the order of A modulo p is greater than p^(1-\epsilon) for all p in a density one subset of the primes. Moreover, the order of A modulo N is greater than N^(1-\epsilon) for all N in a density one subset of the integers.Comment: 12 page

The distribution of spacings between quadratic residues

We study the distribution of spacings between squares modulo q, where q is square-free and highly composite, in the limit as the number of prime factors of q goes to infinity. We show that all correlation functions are Poissonian, which among other things, implies that the spacings between nearest neighbors, normalized to have unit mean, have an exponential distribution.Comment: 38 pages; introduction and section 6.2 revised, references updated. To appear in Duke Math. Journa

On the period of the linear congruential and power generators

We consider the periods of the linear congruential and the power generators modulo $n$ and, for fixed choices of initial parameters, give lower bounds that hold for ``most'' $n$ when $n$ ranges over three different sets: the set of primes, the set of products of two primes (of similar size), and the set of all integers. For most $n$ in these sets, the period is at least $n^{1/2+\epsilon(n)}$ for any monotone function $\epsilon(n)$ tending to zero as $n$ tends to infinity. Assuming the Generalized Riemann Hypothesis, for most $n$ in these sets the period is greater than $n^{1-\epsilon}$ for any $\epsilon >0$. Moreover, the period is unconditionally greater than $n^{1/2+\delta}$, for some fixed $\delta>0$, for a positive proportion of $n$ in the above mentioned sets. These bounds are related to lower bounds on the multiplicative order of an integer $e$ modulo $p-1$, modulo $\lambda(pl)$, and modulo $\lambda(m)$ where $p,l$ range over the primes, $m$ ranges over the integers, and where $\lambda(n)$ is the order of the largest cyclic subgroup of $(\Z/n\Z)^\times$.Comment: 20 pages. One of the quoted results (Theorem 23 in the previous version) is stated for any unbounded monotone function psi(x), but it appears that the proof only supports the case when psi(x) is increasing rather slowly. As a workaround, we provide a modified version of Theorem 23, and change the argument in the proof of Theorem 27 (Theorem 25 in the previous version

Hecke theory and equidistribution for the quantization of linear maps of the torus

We study semi-classical limits of eigenfunctions of a quantized linear hyperbolic automorphism of the torus ("cat map"). For some values of Planck's constant, the spectrum of the quantized map has large degeneracies. Our first goal in this paper is to show that these degeneracies are coupled to the existence of quantum symmetries. There is a commutative group of unitary operators on the state-space which commute with the quantized map and therefore act on its eigenspaces. We call these "Hecke operators", in analogy with the setting of the modular surface. We call the eigenstates of both the quantized map and of all the Hecke operators "Hecke eigenfunctions". Our second goal is to study the semiclassical limit of the Hecke eigenfunctions. We will show that they become equidistributed with respect to Liouville measure, that is the expectation values of quantum observables in these eigenstates converge to the classical phase-space average of the observable.Comment: 37 pages. New title. Spelling mistake in bibliography corrected. To appear in Duke Math.

The fluctuations in the number of points on a hyperelliptic curve over a finite field

The number of points on a hyperelliptic curve over a field of $q$ elements may be expressed as $q+1+S$ where $S$ is a certain character sum. We study fluctuations of $S$ as the curve varies over a large family of hyperelliptic curves of genus $g$. For fixed genus and growing $q$, Katz and Sarnak showed that $S/\sqrt{q}$ is distributed as the trace of a random $2g\times 2g$ unitary symplectic matrix. When the finite field is fixed and the genus grows, we find that the the limiting distribution of $S$ is that of a sum of $q$ independent trinomial random variables taking the values $\pm 1$ with probabilities $1/2(1+q^{-1})$ and the value 0 with probability $1/(q+1)$. When both the genus and the finite field grow, we find that $S/\sqrt{q}$ has a standard Gaussian distribution.Comment: 10 pages. Final versio