  We study how the electron-electron interactions influence the charge
distributions in the metallic state of quasicrystals. As a simple theoretical
model, we introduce an extended Hubbard model on the Penrose lattice, and
numerically solve the model (up to $\sim1.4$ million sites) within the
Hartree-Fock approximation. Because each site on the quasiperiodic lattice has
a different local geometry, the Coulomb interaction, in particular the
intersite one, works in a site-dependent way, leading to a nontrivial
redistribution of the charge. The resultant charge distribution patterns are
not multifractal but characterized by hyperuniformity, which offers a measure
to distinguish various inhomogeneous but ordered distributions. We clarify how
the electron interactions alter the order metric of the hyperuniformity,
revealing that the intersite interaction considerably affects the
hyperuniformity in particular on the electron-rich side.
