A multiparameter variant of the Salem–Zygmund central limit theorem on lacunary trigonometric series
Volume 131 / 2013
Colloquium Mathematicum 131 (2013), 13-27
MSC: Primary 60F15; Secondary 42A55, 42B.
DOI: 10.4064/cm131-1-2
Abstract
We prove the central limit theorem for the multisequence $$ \sum_{1 \leq n_1 \leq N_1} \cdots \sum_{1 \leq n_d \leq N_d} a_{n_1, \ldots ,n_d} \cos (\langle 2\pi \mathbf{m}, A_1^{n_1} \dots A_d^{n_d} \mathbf{x} \rangle) $$ where $\mathbf{m} \in \mathbb Z^s$, $a_{n_1, \ldots ,n_d}$ are reals, $A_1, \ldots ,A_d$ are partially hyperbolic commuting $s\times s$ matrices, and $\mathbf{x}$ is a uniformly distributed random variable in $[0,1]^s$. The main tool is the S-unit theorem.
