A threshold for the best two-term underapproximation by Egyptian fractions

Let $G$ be the greedy algorithm that, for each $\theta \in (0,1]$, produces an infinite sequence of positive integers ${\left(a_n\right)}_{n=1}^{\infty }$ satisfying ${\sum }_{n=1}^{\infty }1/a_n=\theta$. For natural numbers $p<q$, let ${\rm Y}(p,q)$ denote the smallest positive integer $j$ such that $p$ divides $q+j$. Continuing Nathanson’s study of two-term underapproximations, we show that whenever ${\rm Y}(p,q)⩽3$, $G$ gives the (unique) best two-term underapproximation of $p/q$; i.e., if $1/x_1+1/x_2<p/q$ for some $x_1,x_2\in N$, then $1/x_1+1/x_2⩽1/a_1+1/a_2$. However, the same conclusion fails for every ${\rm Y}(p,q)⩾4$. Next, we study stepwise underapproximation by $G$. Let $e_m=\theta -{\sum }_{n=1}^{m}1/a_n$ be the $m$th error term. We compare $1/a_m$ to a superior underapproximation of $e_{m-1}$, denoted by $N/b_m$ ($N\in N_{⩾2}$), and characterize when $1/a_m=N/b_m$. One characterization is $a_{m+1}⩾N{a}_m^2-a_m+1$. Hence, for rational $\theta$, 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.