$\phi_K\circ t_\partial^n$ describes a tight contact structure (on a different 3-manifold) for any $n>N_Y$. Therefore Theorem \[thm1\] is implied by the following.

\[thm2\] For a closed oriented 3-manifold $Y$, there is an integer $N_Y\geq 0$ with the following property: Let $\xi$ be a contact structure on $Y$, and choose any open book decomposition $(S,\phi)$ that supports $\xi$ and has connected binding. Then for any $n>N_Y$, the open book $(S,\phi\circ t^n_{\partial S})$ determines a tight contact structure.

Theorem \[thm1\] follows from Theorem \[thm2\] by observing that open book decompositions for $Y$ are in bijection with those for $-Y$ under a correspondence induced by inverting monodromies. Thus $N_Y$ in the first theorem can be taken as the maximum of $N_Y$ and $N_{-Y}$ from the second.

Theorem \[thm2\] was first observed by [[Ozsváth and Szabó]{}]{} in the case that $Y$ is an $L$-space, in which case $N_Y = 0$ [@Contact Theorem 1.6]. Indeed, in that paper they ask the following question:

[@Contact pg. 43] Given an open book decomposition $(S, \phi)$ for $Y$, what is the minimum $n$ such that $(S,\phi\circ t_{\partial S}^n)$ specifies a tight contact structure?

Theorem \[thm2\] is proved by a generalization of [[Ozsváth and Szabó]{}]{}’s argument, and we obtain 