Suppose $0 <\alpha <1$ is an irrational, and $0 < \gamma_1, \gamma_2 < 1$ are real numbers satisfying $\gamma_i \notin \mathbb{Z} \alpha + \mathbb{Z}$ for $i=1,2$. Consider the sequence $(\left\| k \alpha \right\|)_{k=1}^{\infty}$ of distances from $k\alpha$ to the nearest integer, and denote by $(k_j)$ the subsequence along which $\|k_j \alpha \| \leq 1/k_j$.
My question is the following: Do there exist pairs $(\gamma_1, \gamma_2)$ (still requiring $\gamma_i \notin \mathbb{Z}\alpha + \mathbb{Z}$) for which the limit
$\lim_{j\to \infty} \left\| k_j \gamma_1 \right\|\cdot \|k_j \gamma_2\|$
exists and equals zero? And if the answer is yes, are there any bounds on the rate of convergence?
