Extensions of democracy-like properties for sequences with gaps

TL;DR

This paper extends the concepts related to greedy algorithms for sequences with gaps, exploring conditions under which these extended notions are equivalent to traditional ones.

Contribution

It introduces new notions for sequences with gaps and establishes conditions for their equivalence to classical concepts.

Findings

Identifies conditions on sequences with gaps for equivalence of extended and classical notions.

Analyzes inequalities involving sequence indices to determine equivalence.

Provides theoretical framework for understanding greedy algorithms in the context of sequences with gaps.

Abstract

In \cite{O2015}, T. Oikhberg introduced and studied variants of the greedy and weak greedy algorithms for sequences with gaps. In this paper, we extend some of the notions that appear naturally in connection with these algorithms to the context of sequences with gaps. In particular, we will consider sequences of natural numbers for which the inequality $n_{k+1}\le \C n_{k}$ or $n_{k+1}\le \C+n_k$ holds for a positive constant $\C$ and all $k$, and find conditions under which the extended notions are equivalent their regular counterparts.