  We sharpen a gap theorem of Chan & Lee for nonnegative Ricci curvature
manifolds that have positive asymptotic volume ratio and small enough
scale-invariant integral curvature (so-called "curvature concentration"), by
showing that the curvature concentration need only depend linearly on the
asymptotic volume ratio. We prove the result by exhibiting a long-time Ricci
flow solution with faster than $1/t$ curvature decay, which allows us to shift
the limiting contradiction argument to time infinity and thus obtain an
explicit bound on the size of the gap.
