On matrix elements for the quantized cat map modulo prime powers

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…

Asymptotic density of Catalan numbers modulo 3 and powers of 2

We establish the asymptotic density of the Catalan numbers modulo 3 and modulo powers of 2.

Large supremum norms and small Shannon entropy for Hecke eigenfunctions of quantized cat maps

This paper concerns the behavior of eigenfunctions of quantized cat maps and in particular their supremum norm. We observe that for composite integer values of N, the inverse of Planck's constant, some of the desymmetrized eigenfunctions have very small support and hence very large supremum norm. We also prove an entropy estimate and show that our functions satisfy equality in this estimate. In the case when N is a prime power with even exponent we calculate the supremum norm for a large proportion of all desymmetrized eigenfunctions and we find that for a given N there is essentially at most …

The Catalan numbers have no forbidden residue modulo primes

Let $C_n$ be the $n$th Catalan number. For any prime $p \geq 5$ we show that the set $\{C_n : n \in \mathbb{N} \}$ contains all residues mod $p$. In addition all residues are attained infinitely often. Any positive integer can be expressed as the product of central binomial coefficients modulo $p$. The directed sub-graph of the automata for $C_n \mod p$ consisting of the constant states and transitions between them has a cycle which visits all vertices.

Structure and asymptotics for Catalan numbers modulo primes using automata

Let $C_n$ be the $n$th Catalan number. We show that the asymptotic density of the set $\{n: C_n \equiv 0 \mod p \}$ is $1$ for all primes $p$, We also show that if $n = p^k -1$ then $C_n \equiv -1 \mod p$. Finally we show that if $n \equiv \{ \frac{p+1}{2}, \frac{p+3}{2}, ..., p-2 \} \mod p$ then $p$ divides $C_n$. All results are obtained using the automata method of Rowland and Yassawi.

关于模素数幂的高维猫映射的量子遍历性

On quantum ergodicity for higher dimensional cat maps modulo prime powers

本文针对模整数 $N$ 的高维辛矩阵线性映射量子遍历性模型，利用线性递推序列指数和的界，显式构造了具有幂次偏差节省的模数 $N$ 序列，解决了以往结果无法显式构造“好”模数的问题。