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Michel Waldschmidt (1975/1976)

Séminaire Delange-Pisot-Poitou. Théorie des nombres

The best Diophantine approximation functions by continued fractions

Jing Cheng Tong (1996)

Mathematica Bohemica

Let ξ = [ a 0 ; a 1 , a 2 , , a i , ] be an irrational number in simple continued fraction expansion, p i / q i = [ a 0 ; a 1 , a 2 , , a i ] , M i = q i 2 | ξ - p i / q i | . In this note we find a function G ( R , r ) such that M n + 1 < R and M n - 1 < r imply M n > G ( R , r ) , M n + 1 > R and M n - 1 > r imply M n < G ( R , r ) . Together with a result the author obtained, this shows that to find two best approximation functions H ˜ ( R , r ) and L ˜ ( R , r ) is a well-posed problem. This problem has not been solved yet.

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