DENSITY AND FINITENESS RESULTS ON SUMS OF FRACTIONS

Abstract

We show that the height density of a finite sum of fractions is zero. In fact, we give quantitative estimates in terms of the height function. Then, we focus on the unit fraction solutions in the ring of integers of a given number field. In particular, we prove that finitely many representations of 1 as a sum of unit fractions determines the field of rational numbers among all real number fields. Finally, using non-standard methods, we prove some density and finiteness results on a finite sum of fractions.