[snowwrestler]

This theory of crime actually says that police presence can be safely reduced now, because all those petty crimes are no longer common.

What you're talking about is a theory of how to boot-strap a reduction in crime: start small. But it also implies that once crime is reduced, the reduction will be self-persisting (for the same reason crime was self-persisting--because of the environment). It does not imply that heavy police presence needs to be permanent.

Anyway, it's never been proved in any real way, it's just an idea that got popular. And a whole lot of cities that Giuliani did not lead also saw significant reductions in serious crime over that same period of time.