Rate of convergence of an irrational rotation Let $\alpha, \beta \in \mathbb{R}$. Let $\{x\}$ denote the fractional part of $x$ and let $\|x\| = \min(\{x\}, 1-\{x\})$. 
If we assume that $\alpha$ is irrational, then there exists an increasing sequence of integers $(n_k)_{k \in \mathbb{N}}$ such that $\|n_k \alpha - \beta\| \to 0$. Is it possible that the rate of convergence is exponential, that is, that there exists some $\eta > 1$ such that $\eta^{n_k} \|n_k \alpha - \beta\| \to 0$?
I believe that the fact that the sequence $\{n \alpha\}_{n \in \mathbb{N}}$ is equidistributed implies that if this is true for some $\eta > 1$, then it must be true for every $\eta > 1$, so this seems to be a rather strong condition.
If the rate may be exponential for general $\alpha$ and $\beta$, what about the case when $e^{2\pi i \alpha}$ and $e^{2\pi i \beta}$ are algebraic?
 A: For the algebraic case a lot is known. There is a deep result of Gelfond (see A. O. Gelfond, Transcendental and Algebraic Numbers, Dover, New York, 1960) that if $\lambda$ is an algebraic number of absolute value 1, then for every $\epsilon>0$ there is a constant $C>0$ such that $$|\lambda^n-1|>Ce^{-n\epsilon}$$ for all $n\ge1$. More recent work of Feldman (see Chapter 9 in  Stolarsky's book Algebraic Numbers and Diophantine Approximation, Dekker, 1974) actually shows that there are effectively computable numbers $C=C(\lambda)>0$ and $N=N(\lambda)\ge1$ such that
$$|\lambda^n-1|>Cn^{-N}$$
for all $n\ge1$.
These diophantine results are very useful while investigating the dynamical properties of maps defined algebraically, e.g. automorphisms of tori. See for example my paper "Dynamical Properties of Quasihyperbolic Toral Automorphisms", Ergod. Th. Dynam. Sys. 2 (1982), 49--68.
A: Certainly it is possible for the rate to be exponential: for example, if we define a sequence $(d_n)$ by $d_1:=2$, $d_{n+1}:=2^{d_n}$ then the number
$$\alpha:=\sum_{n=1}^\infty \frac{1}{d_n}$$
satisfies
$$\{d_n\alpha\}=\sum_{k=n+1}^\infty \frac{1}{d_k} \in \left(\frac{1}{d_{n+1}},\frac{2}{d_{n+1}}\right)$$
and we have $d_{n+1}=2^{d_n}$ so the size of this fractional part is exponentially small relative to $d_n$. In the case $\beta=0$ you can construct more general examples using the theory of continued fractions.
