The rate of growth of the denominators in the Oppenheim series
Author:
János Galambos
Journal:
Proc. Amer. Math. Soc. 59 (1976), 913
MSC:
Primary 10K10
MathSciNet review:
0568142
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A BorelCantelli lemma is proved for a sequence of functions of the denominators in the Oppenheim expansion of real numbers. This is then applied to the study of the rate of growth of the denominators in the above series. The laws obtained are almost sure type, that is, valid for (Lebesgue) almost all in the unit interval. The results are new even for the classical expansions of Engel, Sylvester and Cantor (product).
 [1]
Ole
BarndorffNielsen, On the rate of growth of the partial maxima of a
sequence of independent identically distributed random variables,
Math. Scand. 9 (1961), 383–394. MR 0139189
(25 #2625)
 [2]
J.
Galambos, The ergodic properties of the denominators in the
Oppenheim expansion of real numbers into infinite series of rationals,
Quart. J. Math. Oxford Ser. (2) 21 (1970), 177–191.
MR
0258777 (41 #3423)
 [3]
János
Galambos, On infinite series representations of real numbers,
Compositio Math. 27 (1973), 197–204. MR 0332700
(48 #11026)
 [4]
János
Galambos, An iterated logarithm type theorem for the largest
coefficient in continued fractions, Acta Arith. 25
(1973/74), 359–364. MR 0344212
(49 #8952)
 [5]
János
Galambos, Further ergodic results on the Oppenheim series,
Quart. J. Math. Oxford Ser. (2) 25 (1974), 135–141.
MR
0347759 (50 #260)
 [6]
A.
Oppenheim, The representation of real numbers by infinite series of
rationals, Acta Arith. 21 (1972), 391–398. MR 0309877
(46 #8982)
 [7]
Walter
Philipp, A conjecture of Erdős on continued fractions,
Acta Arith. 28 (1975/76), no. 4, 379–386. MR 0387226
(52 #8069)
 [8]
Fritz
Schweiger, Metrische Sätze über
Oppenheimentwicklungen, J. Reine Angew. Math. 254
(1972), 152–159 (German). MR 0297729
(45 #6781)
 [9]
Fritz
Schweiger, Gedämpfte zahlentheoretische Transformationen,
Monatsh. Math. 79 (1975), 67–73 (German, with
English summary). MR 0360500
(50 #12948)
 [10]
W.
Vervaat, Success epochs in Bernoulli trials (with applications in
number theory), Mathematisch Centrum, Amsterdam, 1972. Mathematical
Centre Tracts, No. 42. MR 0328989
(48 #7331)
 [1]
 O. BarndorffNielsen, On the rate of growth of the partial maxima of a sequence of independent identically distributed random variables, Math. Scand. 9 (1961), 383394. MR 25 #2625. MR 0139189 (25:2625)
 [2]
 J. Galambos, The ergodic properties of the denominators in the Oppenheim expansion of real numbers into infinite series of rationals, Quart. J. Math. Oxford Ser. (2) 21 (1970), 177191. MR 41 #3423. MR 0258777 (41:3423)
 [3]
 , On infinite series representations of real numbers, Compositio Math. 27 (1973), 197204. MR 48 #11026. MR 0332700 (48:11026)
 [4]
 , An iterated logarithm type theorem for the largest coefficient in continued fractions, Acta Arith. 25 (1973/74), 359364. MR 49 #8952. MR 0344212 (49:8952)
 [5]
 , Further ergodic results on the Oppenheim series, Quart. J. Math. Oxford Ser. (2) 25 (1974), 135141. MR 50 #260. MR 0347759 (50:260)
 [6]
 A. Oppenheim, The representation of real numbers by infinite series of rationals, Acta Arith. 21 (1972), 391398. MR 46 #8982. MR 0309877 (46:8982)
 [7]
 W. Phillip, A conjecture of Erdös on continued fractions, Acta Arith. 28 (1975/76), 379386. MR 0387226 (52:8069)
 [8]
 F. Schweiger, Metrische Sätze über Oppenheimentwicklungen, J. Reine Angew. Math. 254 (1972), 152159. MR 45 #6781. MR 0297729 (45:6781)
 [9]
 , Gedämpfte zahlentheoretische Transformationen, Monatsh. Math. 79 (1975), 6773. MR 0360500 (50:12948)
 [10]
 W. Vervaat, Success epochs in Bernoulli trials (with applications in number theory), Math. Centre Tracts, no. 42, Math. Centrum, Amsterdam, 1972. MR 48 #7331. MR 0328989 (48:7331)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
10K10
Retrieve articles in all journals
with MSC:
10K10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197605681424
PII:
S 00029939(1976)05681424
Keywords:
Oppenheim series,
denominators,
BorelCantelli lemma,
asymptotic laws,
,
,
Engel series,
Sylvester series,
Cantor product
Article copyright:
© Copyright 1976 American Mathematical Society
