Approximation of real numbers by rationals: Some metric theorems

[摘要] 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. The first result concerns the average rate of approximation: integral(0)(1) rho(n)(x) dx = 3/pi(2) log n/n(2) + O(1/n(2)), n --> infinity. The secund result states that any badly approximable number is better approximable by rationals than all numbers in average. Namely, we show that if x is an element of [0, 1] is a badly approximable number then c(1) less than or equal to n(2) rho(n)(x) less than or equal to c(2) for all integers n greater than or equal to 1 and some constants c(1) > 0, c(2) > 0. The last two theorems can be considered as analogues of Khinchin's metric theorem regarding the behaviour of inferior and superior limits of n(2) rho(n)(x) f(log n), when n --> infinity, for almost all x is an element of [0, 1] and suitable functions f(.). (C) 1996 Academic Press, Inc.

[发布日期] 1996-12-01