Spectrum of weighted Birkhoff average

Let ${s_n}_{ninmathbb{N}}$ be a decreasing nonsummable sequence of positive reals. In this paper, we investigate the weighted Birkhoff average $frac{1}{S_n}sum_{k=0}^{n-1}s_kphi(T^kx)$ on aperiodic irreducible subshift of finite type $Sigma_{bf A}$ where $phi: Sigma_{bf A}mapsto mathbb{R}$ is a continuous potential. Firstly, we show the entropy spectrum of the weighed Birkhoff averages remains the same as that of the Birkhoff averages. Then we prove that the packing spectrum of the weighed Birkhoff averages equals to either that of the Birkhoff averages or the whole space.