On M. Hall's continued fraction theorem

Author:
T. W. Cusick

Journal:
Proc. Amer. Math. Soc. **38** (1973), 253-254

MSC:
Primary 10F20

MathSciNet review:
0309875

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

**[1]**T. W. Cusick,*Sums and products of continued fractions*, Proc. Amer. Math. Soc.**27**(1971), 35–38. MR**0269603**, 10.1090/S0002-9939-1971-0269603-3**[2]**T. W. Cusick and R. A. Lee,*Sums of sets of continued fractions*, Proc. Amer. Math. Soc.**30**(1971), 241–246. MR**0282924**, 10.1090/S0002-9939-1971-0282924-3**[3]**Marshall Hall Jr.,*On the sum and product of continued fractions*, Ann. of Math. (2)**48**(1947), 966–993. MR**0022568**

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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