Bounds on supremum norms for Hecke eigenfunctions of quantized cat maps

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 constant N=1/h, such that the map is diagonalizable (but not upper triangular) modulo N, the Hecke eigenfunctions are uniformly bounded. The purpose of this paper is to show that the same holds for any prime N provided that the map is not upper triangular modulo N. We also find that the supremum norms of Hecke eigenfunctions are << N^epsilon for all epsilon>0 in the case of N square free.Comment: 16 pages. Introduction expanded; comparison with supremum norms of eigenfunctions of the Laplacian added. Bound for square free N adde

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

Variation of the Nazarov-Sodin constant for random plane waves and arithmetic random waves

This is a manuscript containing the full proofs of results announced in [KW], together with some recent updates. We prove that the Nazarov-Sodin constant, which up to a natural scaling gives the leading order growth for the expected number of nodal components of a random Gaussian field, genuinely depends on the field. We then infer the same for "arithmetic random waves", i.e. random toral Laplace eigenfunctions.Comment: 27 pages, 6 figures. To appear in Advances in Mathematic

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