We introduce and study a new complexity function in combinatorics on words, which takes into account the smallest second occurrence time of a factor of an infinite word. We characterize the eventually periodic words and the Sturmian words by means of this function. Then, we establish a new result on repetitions in Sturmian words and show that it is best possible. Let
b
≥
2
b \ge 2
be an integer. We deduce a lower bound for the irrationality exponent of real numbers whose sequence of
b
b
-ary digits is a Sturmian sequence over
{
0
,
1
,
…
,
b
−
1
}
\{0, 1, \ldots , b-1\}
and we prove that this lower bound is best possible. As an application, we derive some information on the
b
b
-ary expansion of
log
(
1
+
1
a
)
\log (1 + \frac {1}{a})
for any integer
a
≥
34
a \ge 34
.