Some grammarians do not like their infinitives split (and this includes me in this case), so you will probably find more not to 's than to not 's. I see no difference in meaning, though I have read threads where posters perceive greater stress on the object when the infinitive is split.
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Anonymousan example of where the "to not" variant makes senseTrue, but I'm moving to France so as not to get fat (or some other rephrasing) would be perfectly fine, and would not split the infinitive.
Davidrock65I want him not to come anywhere closer near my daughter = I don't want him to come anywhere closer near my daughter? Which is more often used?In English the negation is raised to the main clause in almost all cases where the main verb is think or want.
AnonymousWouldn't "I'm moving to France in order not to get fat" be correct as well?Yes.