Approximation of Real Numbers by Rationals: Some Metric Theorems Original Research Article

Letxbe a real number in [0, 1], imagenbe the Farey sequence of ordernandρn(x) be the distance betweenxand imagen. 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 ifxset membership, variant[0, 1] is a badly approximable number thenc1less-than-or-equals, slantn2ρn(x)less-than-or-equals, slantc2for all integersngreater-or-equal, slanted1 and some constantsc1>0,c2>0. The last two theorems can be considered as analogues of Khinchinʹs metric theorem regarding the behaviour of inferior and superior limits ofn2ρn(x) f(log n), whenn→∞, for almost allxset membership, variant[0, 1] and suitable functionsf(·).