Larger Greedy Sums for Reverse Partially Greedy Bases

TL;DR

This paper investigates how enlarging greedy sums affects the properties of reverse partially greedy bases, establishing a characterization that parallels known results for partially greedy bases.

Contribution

It provides a new characterization of reverse partially greedy bases when greedy sums are enlarged, answering a previously posed open question.

Findings

Enlarging greedy sums preserves the reverse partially greedy property.

A characterization of RPG bases under enlarged greedy sums is established.

The result parallels known properties for partially greedy bases.

Abstract

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.