Density and finiteness results on sums of fractions
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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.References
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Additional Information
- Haydar Göral
- Affiliation: Department of Mathematics, Faculty of Sciences, Dokuz Eylül University, Tınaztepe Yerleşkesi, 35390 Buca/İzmir
- Email: hgoral@gmail.com
- Doğa Can Sertbaş
- Affiliation: Department of Mathematics, Faculty of Sciences, Cumhuriyet University, 58140, Sivas, Turkey
- MR Author ID: 137911
- ORCID: 0000-0002-5884-6856
- Email: dogacan.sertbas@gmail.com
- Received by editor(s): November 17, 2017
- Received by editor(s) in revised form: November 26, 2017, May 27, 2018, and June 4, 2018
- Published electronically: November 8, 2018
- Communicated by: Ken Ono
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 567-581
- MSC (2010): Primary 11D68, 03C98
- DOI: https://doi.org/10.1090/proc/14270
- MathSciNet review: 3894896