  Electrons on the lattice subject to a strong magnetic field exhibit the
fractal spectrum of electrons, which is known as the Hofstadter butterfly. In
this work, we investigate unconventional superconductivity in a
three-dimensional Hofstadter butterfly system. While it is generally difficult
to achieve the Hofstadter regime, we show that the quasi-two-dimensional
materials with a tilted magnetic field produce the large-scale superlattices,
which generate the Hofstadter butterfly even at the moderate magnetic field
strength. We first show that the van-Hove singularities of the butterfly flat
bands greatly elevate the superconducting critical temperature, offering a new
mechanism of field-enhanced superconductivity. Furthermore, we demonstrate that
the quantum geometry of the Landau mini-bands plays a crucial role in the
description of the superconductivity, which is shown to be clearly distinct
from the conventional superconductors. Finally, we discuss the relevance of our
results to the recently discovered re-entrant superconductivity of UTe2 in
strong magnetic fields.
