On powers of operators with spectrum in cantor sets and spectral synthesis

For $\xi \in (0,\frac{1}{2})$, let $E_{\xi }$ be the perfect symmetric set associated with $\xi$, that is $$E_{\xi }=\{\exp (2i\pi (1-\xi )\sum _{n=1}^{+\mathrm{\infty }}{ϵ}_n{\xi }^{n-1}):{ϵ}_n=0\text{ or }1(n\ge 1)\}$$ and $$b(\xi )=\frac{\log \frac{1}{\xi }-\log 2}{2\log \frac{1}{\xi }-\log 2}\mathrm{.}$$ Let $q\ge 3$ be an integer and $s$ be a nonnegative real number. We show that any invertible operator $T$ on a Banach space with spectrum contained in $E_{1\mathrm{/}q}$ that satisfies \begin{eqnarray*} & & \big| T^{n} \big| = O \big( n^{s} \big), ,n \rightarrow +\infty \ & $\text{and}$ & \big| T^{-n} \big| = O \big( e^{n^{\beta}} \big), , n \rightarrow +\infty $\text{ for some }$ \beta < b(1/q),\end{eqnarray*} also satisfies the stronger property $\big| T^{-n} \big| = O \big( n^{s} \big), , n \rightarrow +\infty.$ We also show that this result is false for $E_{\xi }$ when $1\mathrm{/}\xi$ is not a Pisot number and that the constant $b(1\mathrm{/}q)$ is sharp. As a consequence we prove that, if $\omega$ is a submulticative weight such that $\omega (n)=(1+n{)}^s,(n\ge 0)$ and $C^{-1} (1+|n|)^s \leq \omega(-n) \leq C e^{n^{\beta}},, (n\geq 0)$, for some constants $C>0$ and $\beta <b(1\mathrm{/}q),$ then $E_{1\mathrm{/}q}$ satisfies spectral synthesis in the Beurling algebra of all continuous functions $f$ on the unit circle $\mathbb{T}$ such that $\sum_{n = -\infty}^{+\infty} | \widehat{f}(n) | \omega (n) < +\infty$.