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

Let $\mathcal{G}$ be the greedy algorithm that, for each $\theta \in (0,1]$, produces an infinite sequence of positive integers $(a_n{)}_{n=1}^{\mathrm{\infty }}$ satisfying ${\sum }_{n=1}^{\mathrm{\infty }}1\mathrm{/}{a}_n=\theta$. For natural numbers $p<q$, let $\mathrm{{\rm Y}}(p,q)$ denote the smallest positive integer $j$ such that $p$ divides $q+j$. Continuing Nathanson’s study of two-term underapproximations, we show that whenever $\mathrm{{\rm Y}}(p,q)⩽3$, $\mathcal{G}$ gives the (unique) best two-term underapproximation of $p\mathrm{/}q$; i.e., if $1\mathrm{/}{x}_1+1\mathrm{/}{x}_2<p\mathrm{/}q$ for some $x_1,x_2\in \mathbb{N}$, then $1\mathrm{/}{x}_1+1\mathrm{/}{x}_2⩽1\mathrm{/}{a}_1+1\mathrm{/}{a}_2$. However, the same conclusion fails for every $\mathrm{{\rm Y}}(p,q)⩾4$. Next, we study stepwise underapproximation by $\mathcal{G}$. Let $e_m=\theta -{\sum }_{n=1}^{m}1\mathrm{/}{a}_n$ be the $m$th error term. We compare $1\mathrm{/}{a}_m$ to a superior underapproximation of $e_{m-1}$, denoted by $N\mathrm{/}{b}_m$ ($N\in {\mathbb{N}}_{⩾2}$), and characterize when $1\mathrm{/}{a}_m=N\mathrm{/}{b}_m$. One characterization is $a_{m+1}⩾N{a}_m^2-a_m+1$. Hence, for rational $\theta$, we only have $1\mathrm{/}{a}_m=N\mathrm{/}{b}_m$ for finitely many $m$. However, there are irrational numbers such that $1\mathrm{/}{a}_m=N\mathrm{/}{b}_m$ for all $m$. Along the way, various auxiliary results are encountered.