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- PÄR KURLBERG
- 1999

1.1. Background. One of the key issues of “Quantum Chaos” is the nature of the semiclassical limit of eigenstates of classically chaotic systems. When the classical system is given by the geodesic flow on a compact Riemannian manifoldM (or rather, on its cotangent bundle), one can formulate the problem as follows: The quantum Hamiltonian is, in suitable… (More)

- PÄR KURLBERG
- 2008

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/ √ q is distributed as the trace of a random 2g×2g unitary… (More)

This sequence was first considered as a pseudorandom number generator by D. H. Lehmer. For the power generator we are given integers e, n > 1 and a seed u = u0 > 1, and we compute the sequence ui+1 = u e i (mod n) so that ui = u ei (mod n). A popular case is e = 2, which is called the Blum–Blum–Shub (BBS) generator. Both of these generators are periodic… (More)

Let K be a number field, let φ ∈ K (t) be a rational map of degree at least 2, and let α, β ∈ K . We show that if α is not in the forward orbit of β, then there is a positive proportion of primes p of K such that α mod p is not in the forward orbit of β mod p. Moreover, we show that a similar result holds for several maps and several points. We also present… (More)

- PÄR KURLBERG
- 2011

For g, n coprime integers, let `g(n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of `g(n) as n ≤ x ranges over integers coprime to g, and x tending to infinity. Assuming the Generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda… (More)

- Laura Fainsilber, Pär Kurlberg, Bernt Wennberg
- SIAM J. Math. Analysis
- 2006

The phase space density f of a dilute gas evolves according to the Boltzmann equation. In the physically relevant case, the gas would be confined to a subset Ω ⊂ R, and then f(x, v, t) : Ω×R3×R+ → R, where x denotes a position in space, v ∈ R is a velocity, and t denotes the time. From a mathematical point of view, it is equally natural to consider the… (More)

Let f1, . . . , fg ∈ C(z) be rational functions, let Φ = (f1, . . . , fg) denote their coordinatewise action on (P1)g, let V ⊂ (P1)g be a proper subvariety, and let P be a point in (P1)g(C). We show that if S = {n > 0 : Φn(P ) ∈ V (C)} does not contain any infinite arithmetic progressions, then S must be a very sparse set of integers. In particular, for any… (More)

- PÄR KURLBERG
- 2006

Abstract. We study extreme values of desymmetrized eigenfunctions (so called Hecke eigenfunctions) for the quantized cat map, a quantization of a hyperbolic linear map of the torus. In a previous paper it was shown that for prime values of the inverse Planck’s constant N = 1/h, such that the map is diagonalizable (but not upper triangular) modulo N , the… (More)

- PÄR KURLBERG
- 1999

We study the distribution of spacings between squares in Z/QZ as the number of prime divisors of Q tends to infinity. In [3] Kurlberg and Rudnick proved that the spacing distribution for square free Q is Poissonian, this paper extends the result to arbitrary Q.

We show that there are sets of integers with asymptotic density arbitrarily close to 1 in which there is no solution to the equation ab = c, with a, b, c in the set. We also consider some natural generalizations, as well as a specific numerical example of a product-free set of integers with asymptotic density greater than 1/2.