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Erschienen in:
2008 | OriginalPaper | Buchkapitel
On the Continued Fraction Expansion of a Class of Numbers
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A classical result of Dirichlet asserts that, for each real number ξ and each real
X ≥
1, there exists a pair of integers (
x
0
, x
1
) satisfying
$$ 1 \leqslant x_0 \leqslant X and \left| {x_0 \xi - x_1 } \right| \leqslant X^{ - 1} $$
(a general reference is Chapter I of [10]). If ξ is irrational, then, by letting
X
tend to infinity, this provides infinitely many rational numbers
x
1
/x
0
with |ξ - x
1
/x
0
≤
x
0
- 2
. By contrast, an irrational real number ξ is said to be
badly approximable
if there exists a constant c
1
> 0 suchthat |ξ -
p/q >
c
1
q
- 2
for each
p/q ∈
ℚ
or,equivalently,if ξ has bounded partial quotients in its continued fraction expansion. Thanks to H. Davenport and W. M. Schmidt, the badly approximable real numbers can also be described as those ξ ∈ ℝ \ ℚ for which the result of Dirichlet can be improved in the sense that there exists a constant c
2
< 1 such that the inequalities 1 ≤ x
0
≤
X and
|x
0
ξ
- x
1
|≤
c
2
X
-1
admit a solution (x
0
, x
1
) ∈ ℤ
2
for each sufficiently large
X
(see Theorem 1 of [2]).