Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On M. Hall's continued fraction theorem


Author: T. W. Cusick
Journal: Proc. Amer. Math. Soc. 38 (1973), 253-254
MSC: Primary 10F20
DOI: https://doi.org/10.1090/S0002-9939-1973-0309875-1
MathSciNet review: 0309875
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For each integer $ k \geqq 2$, let $ F(k)$ denote the set of real numbers $ \alpha $ such that $ 0 \leqq \alpha \leqq 1$ and $ \alpha $ has a continued fraction containing no partial quotient greater than $ k$. A well-known theorem of Marshall Hall, Jr. states that (with the usual definition of a sum of point sets) $ F(4) + F(4)$ contains an interval of length $ \geqq 1$; it follows immediately that every real number is representable as a sum of two real numbers each of which has fractional part in $ F(4)$. In this paper it is shown that every real number is representable as a sum of real numbers each of which has fractional part in $ F(3)$ or $ F(2)$, the number of summands required being 3 or 4, respectively.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 10F20

Retrieve articles in all journals with MSC: 10F20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0309875-1
Keywords: Continued fractions, Cantor sets, sums of sets
Article copyright: © Copyright 1973 American Mathematical Society