  The statistics of recurrence times in broad areas have been reported to obey
universal scaling laws, both for single homogeneous regions (Corral, 2003) and
when averaged over multiple regions (Bak et al.,2002). These unified scaling
laws are characterized by intermediate power law asymptotics. On the other
hand, Molchan (2005) has presented a mathematical proof that, if such a
universal law exists, it is necessarily an exponential, in obvious
contradiction with the data. First, we generalize Molchan's argument to show
that an approximate unified law can be found which is compatible with the
empirical observations when incorporating the impact of the Omori law of
earthquake triggering. We then develop the full theory of the statistics of
inter-event times in the framework of the ETAS model of triggered seismicity
and show that the empirical observations can be fully explained. Our
theoretical expression fits well the empirical statistics over the whole range
of recurrence times, accounting for different regimes by using only the physics
of triggering quantified by Omori's law. The description of the statistics of
recurrence times over multiple regions requires an additional subtle
statistical derivation that maps the fractal geometry of earthquake epicenters
onto the distribution of the average seismic rates in multiple regions. This
yields a prediction in excellent agreement with the empirical data for
reasonable values of the fractal dimension $d \approx 1.8$, the average
clustering ratio $n \approx 0.9$, and the productivity exponent $\alpha \approx
0.9$ times the $b$-value of the Gutenberg-Richter law.
