Underapproximation by Egyptian fractions

Abstract

An increasing sequence $(x_i{)}_{i=1}^n$ of positive integers is an $n$-term Egyptian underapproximation of $\theta \in (0,1]$ if ${\sum }_{i=1}^n\frac{1}{x_i}<\theta$. A greedy algorithm constructs an $n$-term underapproximation of $\theta$. For some but not all numbers $\theta$, the greedy algorithm gives a unique best $n$-term underapproximation 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.

20 pages, minor corrections

Cited by in corpus (2)

• On eventually greedy best underapproximations by Egyptian fractions
• On a conjecture of Erdős and Graham about the Sylvester's sequence