Sato-Tate Equidistribution of Kurlberg-Rudnick Sums-1 Sato-Tate Equidistribution of Kurlberg-Rudnick Sums

Sato-tate Theorems for Finite-field Mellin Transforms

as χ varies over all multiplicative characters of k. For each χ, S(χ) is real, and (by Weil) has absolute value at most 2. Evans found empirically that, for large q = #k, these q − 1 sums were approximately equidistributed for the “Sato-Tate measure” (1/2π) √ 4− x2dx on the closed interval [−2, 2], and asked if this equidistribution was provably true. Rudnick had done numerics on the sums T (χ)...

Sato-tate Theorems for Mellin Transforms over Finite Fields

as χ varies over all multiplicative characters of k. For each χ, S(χ) is real, and (by Weil) has absolute value at most 2. Evans found empirically that, for large q = #k, these q − 1 sums were approximately equidistributed for the “Sato-Tate measure” (1/2π) √ 4− x2dx on the closed interval [−2, 2], and asked if this equidistribution was provably true. Rudnick had done numerics on the sums T (χ)...

On the Fluctuations of Matrix Elements of the Quantum Cat Map

We study the fluctuations of the diagonal matrix elements of the quantum cat map about their limit. We show that after suitable normalization, the fifth centered moment for the Hecke basis vanishes in the semiclassical limit, confirming in part a conjecture of Kurlberg and Rudnick. We also study sums of matrix elements lying in short windows. For observables with zero mean, the first moment of ...

Convolution and Equidistribution: Sato-Tate theorems for finite-field Mellin transforms

The Higher-dimensional Rudnick-kurlberg Conjecture Shamgar Gurevich and Ronny Hadani

In this paper we give a proof of the Hecke quantum unique ergodicity conjecture for the multidimensional Berry-Hannay model. A model of quantum mechanics on the 2n-dimensional torus. This result generalizes the proof of the Rudnick-Kurlberg Conjecture given in [GH3] for the 2-dimensional torus.