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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Dense Egyptian fractions


Author: Greg Martin
Journal: Trans. Amer. Math. Soc. 351 (1999), 3641-3657
MSC (1991): Primary 11D68
Published electronically: March 22, 1999
MathSciNet review: 1608486
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Abstract: Every positive rational number has representations as Egyptian fractions (sums of reciprocals of distinct positive integers) with arbitrarily many terms and with arbitrarily large denominators. However, such representations normally use a very sparse subset of the positive integers up to the largest denominator. We show that for every positive rational there exist representations as Egyptian fractions whose largest denominator is at most $N$ and whose denominators form a positive proportion of the integers up to $N$, for sufficiently large $N$; furthermore, the proportion is within a small factor of best possible.


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Additional Information

Greg Martin
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
Address at time of publication: Department of Mathematics, University of Toronto, Toronto, Canada M5S 3G3
Email: gerg@math.toronto.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02327-2
PII: S 0002-9947(99)02327-2
Received by editor(s): July 7, 1997
Published electronically: March 22, 1999
Article copyright: © Copyright 1999 American Mathematical Society