The Thresholding Greedy Algorithm versus Approximations with Sizes Bounded by Certain Functions ff

Let $X$ be a Banach space and $(e_n)_{n=1}^\infty$ be a basis. For a fixed function $f$ in a certain collection $\mathcal{F}$ (closed under composition), we define and characterize ($f$, greedy) and ($f$, almost greedy) bases. These bases nontrivially extend the classical notion of greedy and almost greedy bases. We study relations among ($f$, (almost) greedy) bases as $f$ varies and show that while a basis is not almost greedy, it can still be ($f$, greedy) for some $f\in \mathcal{F}$. Furthermore, we prove that for all non-identity function $f\in \mathcal{F}$, we have the surprising equivalence \mbox{($f$, greedy)}\ \Longleftrightarrow \ \mbox{($f$, almost greedy)}.Comment: 20 page

Approximation by Egyptian Fractions and the Weak Greedy Algorithm

Let $0 < \theta \leqslant 1$. A sequence of positive integers $(b_n)_{n=1}^\infty$ is called a weak greedy approximation of $\theta$ if $\sum_{n=1}^{\infty}1/b_n = \theta$. We introduce the weak greedy approximation algorithm (WGAA), which, for each $\theta$, produces two sequences of positive integers $(a_n)$ and $(b_n)$ such that a) $\sum_{n=1}^\infty 1/b_n = \theta$; b) $1/a_{n+1} < \theta - \sum_{i=1}^{n}1/b_i < 1/(a_{n+1}-1)$ for all $n\geqslant 1$; c) there exists $t\geqslant 1$ such that $b_n/a_n \leqslant t$ infinitely often. We then investigate when a given weak greedy approximation $(b_n)$ can be produced by the WGAA. Furthermore, we show that for any non-decreasing $(a_n)$ with $a_1\geqslant 2$ and $a_n\rightarrow\infty$, there exist $\theta$ and $(b_n)$ such that a) and b) are satisfied; whether c) is also satisfied depends on the sequence $(a_n)$. Finally, we address the uniqueness of $\theta$ and $(b_n)$ and apply our framework to specific sequences.Comment: 14 pages, to appear in Indag. Math. (N.S.

When the Nontrivial, Small Divisors of a Natural Number are in Arithmetic Progression

Iannucci considered the positive divisors of a natural number $n$ that do not exceed $\sqrt{n}$ and found all forms of numbers whose such divisors are in arithmetic progression. In this paper, we generalize Iannucci's result by excluding the trivial divisors $1$ and $\sqrt{n}$ (when $n$ is a square). Surprisingly, the length of our arithmetic progression cannot exceed $5$.Comment: 8 pages. Edited according to anonymous referees' suggestions. To appear in Quaestiones Mathematica