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

 

Results on sums of continued fractions


Author: James L. Hlavka
Journal: Trans. Amer. Math. Soc. 211 (1975), 123-134
MSC: Primary 10F20
MathSciNet review: 0376545
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Abstract: Let $ F(m)$ be the (Cantor) set of infinite continued fractions with partial quotients no greater than m and let $ F(m) + F(n) = \{ \alpha + \beta :\alpha \in F(m),\beta \in F(n)\} $. We show that $ F(3) + F(4)$ is an interval of length 1.14 ... so every real number is the sum of an integer, an element of $ F(3)$ and an element of $ F(4)$. Similar results are given for $ F(2) + F(7),F(2) + F(2) + F(4),F(2) + F(3) + F(3)$ and $ F(2) + F(2) + F(2) + F(2)$. The techniques used are applicable to any Cantor sets in R for which certain parameters can be evaluated.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0376545-X
PII: S 0002-9947(1975)0376545-X
Keywords: Continued fractions, sums of Cantor sets, congruence modulo 1
Article copyright: © Copyright 1975 American Mathematical Society