Approximation by Several Rationals

TL;DR

This paper improves approximation results for a rational fraction by sums of multiple smaller rational fractions, specifically for cases with three and four terms, using a novel approach.

Contribution

It introduces a new method to enhance approximation bounds for sums of several rational fractions in specific cases n=3 and n=4.

Findings

Improved approximation bounds for n=3 and n=4 cases.

New approach yields better results than previous methods.

Applicable to certain ranges of denominators q_1,..., q_n.

Abstract

Following T. H. Chan, we consider the problem of approximation of a given rational fraction a/q by sums of several rational fractions a_1/q_1, ..., a_n/q_n with smaller denominators. We show that in the special cases of n=3 and n=4 and certain admissible ranges for the denominators q_1,..., q_n, one can improve a result of T. H. Chan by using a different approach.