Let
x
x
be a real irrational number and
p
n
/
q
n
,
n
=
0
,
1
,
2
,
…
p_n/q_n, \,\,n=0,1,2,\ldots
its sequence of continued fraction convergents. Define
θ
n
=
q
n
|
q
n
x
−
p
n
|
.
\theta _n=q_n\big |q_nx-p_n\big |.
For almost all
x
x
the distribution function of the sequence
|
θ
n
+
1
−
θ
n
−
1
|
,
n
=
0
,
1
,
2
,
…
\big |\theta _{n+1}-\theta _{n-1}\big |, \,\,n=0,1,2,\ldots
is determined and its expectation calculated.