Does there exist $\theta\in\mathbb{T}$ such that $|V(\theta+m\alpha)-V(\theta+n\alpha)|\geq\frac{\gamma}{|n-m|^{\tau}}$

Suppose $\alpha$ is Diophantine, i.e., there exist $\gamma, \tau$ such that $\|n\alpha\|_{\mathbb T }\geq\frac{\gamma}{|n|^\tau}$. For any analytic function $V(\cdot)\in C^\omega(\mathbb{T},\mathbb{R})$, does there exist some $\theta\in\mathbb{T}$ and $\tau_1, \gamma_1$ such that $|V(\theta+m\alpha)-V(\theta+n\alpha)|\geq\frac{\gamma_1}{|n-m|^{\tau_1}}$ for any $m, n\in\mathbb{Z}$?

I only know when $m$ is fixed, there exists a positive measure $\theta$ such that the above inequality holds, how to show the inequality for any $m, n$?