Asymptotic Distribution of the Distance Function to the Farey Points Original Research Article

Letxbe a real number in [0, 1], imagenbe the Farey sequence of ordernandρn(x) be the distance betweenxand imagen. Assuming thatn→∞ we derive the asymptotic distributions of the functionsn2ρn(x) andnρn(x′/n), 0less-than-or-equals, slantx′less-than-or-equals, slantn. We also establish the asymptotics for ∫10 ρδn(x) dx, for all realδ.