  Renormalization group calculations for fluctuation-dominated
reaction-diffusion systems are generally in agreement with simulations and
exact solutions. However, simulations of the single-species reactions
3A->(0,A,2A) at their upper critical dimension d_c=1 have found asymptotic
densities argued to be inconsistent with renormalization group predictions. We
show that this discrepancy is resolved by inclusion of the leading corrections
to scaling, which we derive explicitly and show to be universal, a property not
shared by the A+A->(0,A) reactions. Finally, we demonstrate that two previous
Smoluchowski approaches to this problem reduce, with various corrections, to a
single theory which yields, surprisingly, the same asymptotic density as the
renormalization group.
