DENSITY AND FINITENESS RESULTS ON SUMS OF FRACTIONS

Goral H., Sertbas D. C.

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, vol.147, no.2, pp.567-581, 2019 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 147 Issue: 2
  • Publication Date: 2019
  • Doi Number: 10.1090/proc/14270
  • Journal Name: PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.567-581
  • Keywords: Egyptian fractions, height function, quantitative estimates, number fields, nonstandard analysis, NUMBER
  • Çukurova University Affiliated: No

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.