An asymptotic equivalence between two frame perturbation theorems

In this paper, two stability results regarding exponential frames are compared. The theorems, (one proven herein, and the other in \cite{SZ}), each give a constant such that if $\sup_{n \in \mathbb{Z^d}}\| \epsilon_n \|_\infty < C$, and $(e^{i \langle \cdot , t_n \rangle})_{n \in \mathbb{Z}^d}$ is a frame for $L_2[-\pi,\pi]^d$, then $(e^{i \langle \cdot , t_n +\epsilon_n \rangle})_{n \in \mathbb{Z}^d}$ is a frame for $L_2[-\pi,\pi]^d$. These two constants are shown to be asymptotically equivalent for large values of $d$.