## On M. Hall’s continued fraction theorem

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- by T. W. Cusick
- Proc. Amer. Math. Soc.
**38**(1973), 253-254 - DOI: https://doi.org/10.1090/S0002-9939-1973-0309875-1
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## 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

- 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

## Bibliographic Information

- © Copyright 1973 American Mathematical Society
- 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