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

Let G be the greedy algorithm that, for each θ ∈ (0 , 1 ] , produces an infinite sequence of positive integers (a n) n = 1 ∞ satisfying ∑ n = 1 ∞ 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) ⩽ 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 ∈ N , then 1 / x 1 + 1 / x 2 ⩽ 1 / a 1 + 1 / a 2 . However, the same conclusion fails for every Υ (p , q) ⩾ 4. Next, we study stepwise underapproximation by G. Let e m = θ − ∑ 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 ∈ 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 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. [ABSTRACT FROM AUTHOR]