Approximation of Real Numbers by Rationals: Some Metric Theorems
✍ Scribed by Pavel Kargaev; Anatoly Zhigljavsky
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 591 KB
- Volume
- 61
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
✦ Synopsis
Let x be a real number in [0, 1], F n be the Farey sequence of order n and \ n (x) be the distance between x and F n . The first result concerns the average rate of approximation:
The second result states that any badly approximable number is better approximable by rationals than all numbers in average. Namely, we show that if x # [0, 1] is a badly approximable number then c 1 n 2 \ n (x) c 2 for all integers n 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 \ n (x) f (log n), when n Ä , for almost all x # [0, 1] and suitable functions f ( } ).