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By what you say you have already proved, $\root3\of d$ badly approximable implies $d^{-1/3}$ badly approximable implies $dd^{-1/3}$ badly approximable, but there's your $d^{2/3}$.


$F$ is a complex-valued function, so you can think of "peaks" as local maxima of $|F(\rho e(\alpha))|$, where $0 < \rho < 1$ is fixed. You can imagine that these peaks happen around certain rational values $\alpha = a/q$ ($q$ not too large) because the oscillations of $e(\alpha)$ are "in phase" with each other at rational points of the $[0,1)$ ...


(I rename $a_n$ as $c_n$) the minimal polynomial of an algebraic number $\alpha = \alpha_1$ is $$P(x) = \sum_{k=0}^n c_k x^k = c_n \prod_{j=1}^n (x-\alpha_j)$$ where by definition of an algebraic number, $P \in \mathbb{Z}[x]$ : the coefficients $c_k \in \mathbb{Z}$, and is irreducible over $\mathbb{Q}[x]$ : the roots $\alpha_j \not \in \mathbb{Q}$. for ...

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