They could try the 3x rental value thing on you as a "non-resident" - though I'm not entirely sure how that works.

Where you'll come onto their radar is at the close of the purchase of the property. They will need to have some indication of your "primary residence" so they'll know where to send the taxes foncières and taxe d'habitation bills if not to the house you buy. The notaires are responsible for setting up that sort of thing, so my guess is that you'll hit their radar at that point.

Giant Panda's warning about the inheritance laws here is worth considering, too. Any house you own in France is subject to French inheritance laws, no matter what. Consult a good notaire who can explain your options to you. (Or you can take my husband's approach, which is "heck, after I'm dead, I really won't care one way or another, will I?" )
Cheers,
Bev