We have an algebraic number $a$ and a real number $b$. Can the following inequality have infinitely many solutions for $n \in \mathbb{N}$?
$$ \{an\} \in [b - \frac{1}{2^n}, b + \frac{1}{2^n}] $$
Where ${x}$ denotes the fractional part of x, ${x} = x - [x]$.
Background. I encountered this problem when I was dealing with some kronecker approximations, I wanted to show that for a given point $(b_1t \mod 2\pi, b_2t \mod 2\pi)$ where $b_1,b_2$ are algebraic numbers who are linearly independent over rational numbers, one can not get exponentially close to a given fixed point $(\phi_1,\phi_2) \in [-1,1]^2$. Some special cases of this problem could be solved by Baker but I was unable to solve it in the general case.