Let $\theta\in [0,1]\setminus\mathbb{Q}$. Let $\alpha_0=\theta$ and $\alpha_1=1$. Let $0<p_0<1$ and $p_1=1-p_0$. For a finite word $I=(i_1, i_2, \dots, i_n)\in \{0,1\}^n$, denote by $I'=(i_1, i_2, \dots, i_{n-1})$, $\alpha_I=\alpha_{i_1}+\alpha_{i_2}+\dots+\alpha_{i_n}$ and $P_I=p_{i_1}p_{i_2}\dots p_{i_n}$. For $R>0$, define $$ T(R):=\{I\in \cup_n\{0,1\}^n: \alpha_I\ge R, \alpha_{I'}< R \}. $$
Let $$ H_{\theta}(R)=\sum_{I\in T(R)}p_I e^{2\pi i \alpha_I}. $$ I want to know
(1) Does $H_{\theta}(R)$ tend to zero as $R$ tends to $+\infty$?
(2) If the first question has the positive answer, what is the rate of convergence?
For question (1), I think renewal theory may be helpful, but I don't know. For question (2), I think the diophantine qpproximation of $\theta$ may affect the rate of convergence.
