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A.P.
A.P.
20:56
@Michael, I found a partial answer to my conjecture in this paper, theorem 3, pages 5-6.
In the slightly simpler case where $\beta > 1$ is a real algebraic integer, this result tells us that if
$$ \left( \sum_{k = 1}^n \log(\max(1,|\beta^{(k)}|) \right) \frac{1}{\log |\beta|} < 1 + \frac{1}{c} $$
for some constant $c > 0$, where $\beta^{(1)}=\beta,\dotsc,\beta^{(n)}$ are all the conjugates of $\beta$, then
for any given $\alpha \in [0,1]$ algebraic, the complexity function $p_{\alpha}$ of its (greedy) $\beta$-expansion is either bounded or it satisfies
$$ \liminf_{n \to \infty} \frac{p_{\alpha}(n)}{n} > c$$

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