We consider one dimensional quantum Ising spin-1/2 chains with two-valued nearest neighbor couplings arranged in a quasiperiodic sequence, with uniform, transverse magnetic field. By employing the transfer matrix technique and investigating the dynamics of the corresponding trace map, we show that in the thermodynamic limit the energy spectrum is a Cantor set of zero Lebesgue measure. Moreover, we show that local Hausdorff dimension is continuous and nonconstant over the spectrum. This forms the rigorous counterpart of numerous numerical studies. We also show that the box-counting and the Hausdorff dimensions (both local and global) coincide.