Larger Greedy Sums for Reverse Partially Greedy Bases

An interesting result due to Dilworth et al. was that if we enlarge greedy sums by a constant factor $\lambda > 1$ in the condition defining the greedy property, then we obtain an equivalence of the almost greedy property, a strictly weaker property. Previously, the author of the present paper showed that enlarging greedy sums by $\lambda$ in the condition defining the partially greedy (PG) property also strictly weakens the property. However, enlarging greedy sums in the definition of reverse partially greedy (RPG) bases by Dilworth and Khurana again gives RPG bases. The companion of PG and RPG bases suggests the existence of a characterization of RPG bases which, when greedy sums are enlarged, gives an analog of a result that holds for partially greedy bases. In this paper, we show that such a characterization indeed exists, answering positively a question previously posed by the author.