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

Abstract

Let G be the greedy algorithm that, for each θ∈(0,1], produces an infinite sequence of positive integers (an)n=1∞ satisfying ∑n=1∞1/an=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)⩽3, G gives the (unique) best two-term underapproximation of p/q; i.e., if 1/x1+1/x2<p/q for some x1,x2∈N, then 1/x1+1/x2⩽1/a1+1/a2. However, the same conclusion fails for every Υ(p,q)⩾4. Next, we study stepwise underapproximation by G. Let em=θ−∑n=1m1/an be the mth error term. We compare 1/am to a superior underapproximation of em−1, denoted by N/bm (N∈N⩾2), and characterize when 1/am=N/bm. One characterization is am+1⩾Nam2−am+1. Hence, for rational θ, we only have 1/am=N/bm for finitely many m. However, there are irrational numbers such that 1/am=N/bm for all m. Along the way, various auxiliary results are encountered.