A recurrence property of smooth functions

Let P and Q be real polynomials of degrees d and e, respectively, and f a periodic function. It is shown that, if f is s times differentiable at Q(0), where s≧7 de ³ log 14 e ³, then for every e{open}>0 the diophantine inequality ≧FF5C;P(x)f(Q(x)) -P(0)f(Q(0)) -y≧ εx≠0, has a solution. This settles in particular a question raised by Furstenberg and Weiss [6].