By a classical result of Gauss and Kuzmin, the continued
fraction expansion of a “random” real number contains each digit
with asymptotic
frequency
.
We generalize this result in two directions: First, for certain sets
, we
establish simple explicit formulas for the frequency with which the continued
fraction expansion of a random real number contains a digit from the set
. For example, we show
that digits of the form
,
where
is prime,
appear with frequency
.
Second, we obtain a simple formula for the frequency with which a string of
consecutive
digits
appears in the continued fraction expansion of a random real number. In particular, when
, this frequency
is given by
,
where
is the
-th
Fibonacci number.
Finally, we compare the frequencies predicted by these results with actual
frequencies found among the first 300 million continued fraction digits of
, and
we provide strong statistical evidence that the continued fraction expansion of
behaves like that of a random real number.