Approximation of Real Numbers by Rationals: Some Metric Theorems

AbstractLetxbe a real number in [0, 1], Fnbe the Farey sequence of ordernandρn(x) be the distance betweenxand Fn. The first result concerns the average rate of approximation:[formula]The second result states that any badly approximable number is better approximable by rationals than all numbers in average. Namely, we show that ifx∈[0, 1] is a badly approximable number thenc1⩽n2ρn(x)⩽c2for all integersn⩾1 and some constantsc1>0,c2>0. The last two theorems can be considered as analogues of Khinchin's metric theorem regarding the behaviour of inferior and superior limits ofn2ρn(x)f(logn), whenn→∞, for almost allx∈[0, 1] and suitable functionsf(·)