For Ī¾ā(0,21ā), let EĪ¾ā be the perfect
symmetric set associated with Ī¾, that is EĪ¾ā={exp(2iĻ(1āĪ¾)n=1ā+āāĻµnāĪ¾nā1):Ļµnā=0Ā orĀ 1(nā„1)} and b(Ī¾)=2logĪ¾1āālog2logĪ¾1āālog2ā. Let
qā„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 E1/qā
that satisfies \begin{eqnarray*} & & \big\| T^{n} \big\| = O \big( n^{s} \big),
\,n \rightarrow +\infty \\ & \textrm{and} & \big\| T^{-n} \big\| = O \big(
e^{n^{\beta}} \big), \, n \rightarrow +\infty \textrm{ for some } \beta <
b(1/q),\end{eqnarray*} also satisfies the stronger property āTānā=O(ns),nā+ā. We also show that this
result is false for EĪ¾ā when 1/Ī¾ is not a Pisot number and that the
constant b(1/q) is sharp. As a consequence we prove that, if Ļ is a
submulticative weight such that Ļ(n)=(1+n)s,(nā„0) and Cā1(1+ā£nā£)sā¤Ļ(ān)ā¤CenĪ²,(nā„0), for some
constants C>0 and Ī²<b(1/q), then E1/qā satisfies spectral
synthesis in the Beurling algebra of all continuous functions f on the unit
circle T such that ān=āā+āāā£fā(n)ā£Ļ(n)<+ā