Underapproximation by Egyptian fractions

Let 0 < theta <= 1. An increasing sequence (xi)ni=1 of positive integers is an n-term Egyptian underapproximation sequence of theta if Sigma n xi < theta. A greedy algorithm constructs an n-1 i=1 term underapproximation sequence of theta. For some but not all numbers theta, the greedy algorithm gives a unique best n-term underapproximation sequence for all n. An infinite set of rational numbers is constructed for which the greedy underapproximations are best, and numbers for which the greedy algorithm is not best are also studied.(c) 2022 Elsevier Inc. All rights reserved.