A Threshold for the Best Two-term Underapproximation by Egyptian Fractions

Abstract

Let $\mathcal{G}$ be the greedy algorithm that, for each $θ\in (0,1]$, produces an infinite sequence of positive integers $(a_n)_{n=1}^\infty$ satisfying $\sum_{n=1}^\infty 1/a_n = θ$. For natural numbers $p < q$, let $Υ(p,q)$ denote the smallest positive integer $j$ such that $p$ divides $q+j$. Continuing Nathanson's study of two-term underapproximations, we show that whenever $Υ(p,q) \leqslant 3$, $\mathcal{G}$ gives the (unique) best two-term underapproximation of $p/q$; i.e., if $1/x_1 + 1/x_2 < p/q$ for some $x_1, x_2\in \mathbb{N}$, then $1/x_1 + 1/x_2 \leqslant 1/a_1+1/a_2$. However, the same conclusion fails for every $Υ(p,q)\geqslant 4$. Next, we study stepwise underapproximation by $\mathcal{G}$. Let $e_{m} = θ- \sum_{n=1}^{m}1/a_n$ be the $m$th error term. We compare $1/a_m$ to a superior underapproximation of $e_{m-1}$, denoted by $N/b_m$ ($N \in\mathbb{N}_{\geqslant 2}$), and characterize when $1/a_m = N/b_m$. One characterization is $a_{m+1} \geqslant N a_m^2 - a_m + 1$. Hence, for rational $θ$, we only have $1/a_m = N/b_m$ for finitely many $m$. However, there are irrational numbers such that $1/a_m = N/b_m$ for all $m$. Along the way, various auxiliary results are encountered.

Figures (2)

• Figure 1: Graph of $\Phi(x)$ for $x\in [1/10,1]$. The set of discontinuities of $\Phi$ is $D = \{1/n: n\in \mathbb{N}_{\geqslant 2}\}$. In particular, for $n\in\mathbb{N}_{\geqslant 2}$, $\lim_{\theta\rightarrow (1/n)^+}\Phi(\theta) = +\infty$, while $\lim_{\theta\rightarrow (1/n)^-}\Phi(\theta) \ =\ 1^+$.
• Figure 2: The two bounds for $\ell$ are strictly between $x_1$ and $x_2$ for any value of $s$.

Theorems & Definitions (56)

• Example 1.1
• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Proposition 1.6
• Corollary 1.7
• proof
• Corollary 1.8
• proof