Bibliography
B. H. Bissinger, A generalization of continued fractions,Bulletin of the Amer. Math. Soc.,50 (1944), pp. 868–876.
C. I. Everett, Representations for real numbers,Bulletin of the Amer. Math. Soc.,52 (1946), pp. 861–869.
W. Bolyai,Tentamen iuventutem studiosam in elementa matheseos purae elementaris ac sublimioris methodo intuitiva evidentiaque huic propria introducendi, ed. sec. (Budapest, 1897), Vol. I.
Gy. Farkas, A Bolyai-féle algoritmus,Értekezések a matematikai tudományok köréből,8 (1881), pp. 1–8, further seeP. Veres, A Bolyai-féle algoritmus,Mennyiségtani és Term. Tud. Didaktikai Lapok,1 (1943), pp. 57–62.
É. Borel, Les probabilités dénombrables et leurs applications arithmétiques,Rendiconti del Circ. Mat. di Palermo,27 (1909), pp. 247–271.
D. Raikoff, On some arithmetical properties of summable functions,Mat. Sbornik,1 (1936), pp. 377–384.
R. O. Kuzmin, Sur un problème de Gauss,Atti del Congresso Internazionale del Matematici Bologna, (1928), Vol. VI, pp. 83–89.
P. Lévy,Théorie de l'addition des variables aléatoires (Paris, 1954), Ch. IX, pp. 290.
A. Khintchine, Metrische Kettenbruchprobleme,Comp. Math.,1 (1935), pp. 359–382.
A. Khintchine, Zur metrischen Kettenbruchtheorie,Comp. Math.,3 (1936), pp. 276–285.
A. Khintchine,Kettenbrüche (Leipzig, 1956).
C. Ryll-Nardzewski, On the ergodic theorems. II. Ergodic theory of continued fractions,Studia Math.,12 (1951), pp. 74–79.
S. Hartman E. Marczewski C. Ryll-Nardzewski, Théorèmes ergodiques et leurs applications,Coll. Math.,2 (1951), pp. 109–123.
S. Hartman, Quelques propriétés ergodiques des fractions continues,Studia Math.,12 (1951), pp. 271–278.
F. Riesz, Sur la théorie ergodique,Commentarii Math. Helv.,1 (1944–45), pp. 221–239.
A. Rényi, Valós számok előállítására szolgáló algoritmusokról,MTA Mat. és. Fiz. Oszt. Közl.,7 (1957), pp. 265–293.
N. Dunford andD. S. Miller, On the ergodic theorem,Trans. Amer. Math. Soc.,60 (1946), pp. 538–549.
F. Riesz, On a recent generalization of G. D. Birkhoff's ergodic theorem,Acta Sci. Math. Szeged,11 (1948), pp. 193–200.
K. Knopp, Mengentheoretische Behandlung einiger. Probleme der diophantischen Approximationen und der transfiniten. Wahrscheinlichkeiten,Math. Annalen,95 (1926), pp. 409–426.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rényi, A. Representations for real numbers and their ergodic properties. Acta Mathematica Academiae Scientiarum Hungaricae 8, 477–493 (1957). https://doi.org/10.1007/BF02020331
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02020331