## 15. ÖMG-Kongress

Jahrestagung der Deutschen Mathematikervereinigung

#### 16. bis 22. September 2001 in Wien

**Sektion 2 - Zahlentheorie**

Montag, 17. September 2001, 15.00, Hörsaal 41

**Some Optimal Error Bounds in the Metric Theory of f-Expansions and Diophantine Approximations **

**Lothar Heinrich**,

**Universität Augsburg**

In the first part we consider partial sums
where
the positve integers
are defined by the expansion
of a real number
which is chosen according to some probability measure .
Under some regularity conditions imposed on the strictly monotone
function
we prove that the suitably normalized
sum
has an stable limit distribution
(
) and derive uniform bounds of the approximation
error provided possesses a strictly positive,
Lipschitz continuous Lebesgue density. Special emphasis is put on
the case
for which
coincides with the
power sum
of the partial
quotients
and
for (where
)
of the continuous fraction expansion
.
In the second part we present large deviation relations
(in the sense of H. Cramér) for the sequences
and
, where ( resp.
) is the
denominator (resp. denumerator) of the th approximant
of
. 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 transformations.
Math. Nachr. **182**, 185 - 214. |

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