Asymptotic distribution of the distance function to the Farey points

Let x be a real number in [0, 1], F-n be the Farey sequence of order n and rho(n)(x) be the distance between x and F-n. Assuming that n --> infinity we derive the asymptotic distributions of the functions n(2) rho(n)(x) and n rho(n)(x'/n), 0 less than or equal to x' less than or equal to n. We also establish the asymptotics for integral(0)(1) rho(n)(delta))(x) dx, for all real delta. (C) 1997 Academic Press.