Results on sums of continued fractions
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- by James L. Hlavka PDF
- Trans. Amer. Math. Soc. 211 (1975), 123-134 Request permission
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.References
- T. W. Cusick, Sums and products of continued fractions, Proc. Amer. Math. Soc. 27 (1971), 35–38. MR 269603, DOI 10.1090/S0002-9939-1971-0269603-3
- T. W. Cusick and R. A. Lee, Sums of sets of continued fractions, Proc. Amer. Math. Soc. 30 (1971), 241–246. MR 282924, DOI 10.1090/S0002-9939-1971-0282924-3
- Marshall Hall Jr., On the sum and product of continued fractions, Ann. of Math. (2) 48 (1947), 966–993. MR 22568, DOI 10.2307/1969389
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 211 (1975), 123-134
- MSC: Primary 10F20
- DOI: https://doi.org/10.1090/S0002-9947-1975-0376545-X
- MathSciNet review: 0376545