Redaktion: K. Sigmund, G. Greschonig (Univ. Wien, Strudlhofgasse ...

68 Zahlentheorie
modulo p. Dabei treten die p-adischen Ziffernentwicklungen
k�
von n und k auf. Die Kongruenz von Lucas wurde in [1] verwendet, um die Verteilung
� von n
in den Restklassen modulo p zu studieren. Ein Resultat von Granville
k�
[2] ermöglicht es nun, dieses Resultat auf Restklassen modulo Primzahlpotenzen
auszudehnen. Dazu werden gewisse Ziffernfunktionen asymptotisch untersucht,
die von Ziffernblocks abhängen.
Restklasse von � n
[1] D. Barbolosi and P. J. Grabner, Distribution des coefficients multinomiaux
et q-binomiaux modulo p, Indag. Math. 7 (1996), 129–135
[2] A. Granville, Arithmetic properties of binomial coefficients. I. Binomial coefficients
modulo prime powers, Organic Mathematics (Burnaby, BC, 1995)
(J. Borwein, P. Borwein, L. Jörgenson, and R. Corless, eds.), Amer. Math.
Soc., Providence, RI, 1997, pp. 253–276
Some Optimal Error Bounds in the Metric Theory of
f-Expansions and Diophantine Approximations
LOTHAR HEINRICH
Institut für Mathematik, Universität Augsburg,
Universitätsstr. 14, D-86135 Augsburg
heinrich@math.uni-augsburg.de
http://www.math.uni-augsburg.de/stochastik/heinrich
In the first part we consider S� partial sums α� �
� � � �
n
ξ1����� f f 1� α �
�
�
� �����
f ξn����� 1� α where the positve ξ1� ξ2������� integers are defined by the � f � expansion
�
which is chosen according to some
1� 0�
f � � ξ1 f � � ����� ��� ξ2 of a real number � ω
probability � measure . Under some regularity conditions imposed on the strictly
monotone function f : � ∞����
a�
S� α� �
n � f has an � α stable limit distribution � (0 � α 2) and derive uniform bounds
of the approximation error � provided possesses a strictly positive, Lipschitz
� 0� 1� we prove that the suitably normalized sum
continuous Lebesgue density. Special emphasis is put on the case f � 1� x x� �
S�
1� p� �
for which �
p
n f coincides with the power sum ξ1 � ξ ����� � p �
� 1� 2� n p
�
of
the partial ω� � �
� 1� ω� quotients ξ1
ω� � �
and ξk ξ1 � T ω� k 1 � for k 2 (where
T ω � 1� ω � ω� ) of the continuous fraction expansion ω � 1� � ξ2������� � . In the
ξ1� �
second part we present large deviation relations (in the sense of H. Cramér) � for
the ω� sequences � logqn � and log ω � � �
pn ω��� qn , where qn ( resp. pn ) is ω��� the
denominator (resp. denumerator) of the n� pn� th approximant qn � ξn� of � � ξ1���������
0� 1� ω . The proofs of the results are based on a method developed in [1].
�
[1] HEINRICH, L. (1996) Mixing properties and central limit theorem for a class
of non-identical piecewise monotonic C2� transformations. Math. Nachr.
182, 185 - 214.