On M. Hall's continued fraction theorem
Author:
T. W. Cusick
Journal:
Proc. Amer. Math. Soc. 38 (1973), 253254
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 wellknown 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
(42 #4498), http://dx.doi.org/10.1090/S00029939197102696033
 [2]
T.
W. Cusick and R.
A. Lee, Sums of sets of continued
fractions, Proc. Amer. Math. Soc. 30 (1971), 241–246. MR 0282924
(44 #158), http://dx.doi.org/10.1090/S00029939197102829243
 [3]
Marshall
Hall Jr., On the sum and product of continued fractions, Ann.
of Math. (2) 48 (1947), 966–993. MR 0022568
(9,226b)
 [1]
 T. W. Cusick, Sums and products of continued fractions, Proc. Amer. Math. Soc. 27 (1971), 3538. MR 42 #4498. MR 0269603 (42:4498)
 [2]
 T. W. Cusick and R. A. Lee, Sums of sets of continued fractions, Proc. Amer. Math. Soc. 30 (1971), 241246. MR 0282924 (44:158)
 [3]
 Marshall Hall, Jr., On the sum and product of continued fractions, Ann. of Math. (2) 48 (1947), 966993. MR 9, 226. MR 0022568 (9:226b)
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DOI:
http://dx.doi.org/10.1090/S00029939197303098751
PII:
S 00029939(1973)03098751
Keywords:
Continued fractions,
Cantor sets,
sums of sets
Article copyright:
© Copyright 1973
American Mathematical Society
