On M. Hall’s continued fraction theorem

Cusick, T. W.

doi:10.1090/S0002-9939-1973-0309875-1

On M. Hall’s continued fraction theorem
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Proc. Amer. Math. Soc. 38 (1973), 253-254 Request permission

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