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| SYMMETRIC AND ASYMMETRIC DIOPHANTINE APPROXIMATION |
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Citation： |
TONG Jingcheng.SYMMETRIC AND ASYMMETRIC DIOPHANTINE APPROXIMATION[J].Chinese Annals of Mathematics B,2004,25(1):139~142 |
| Page view： 1231
Net amount： 701 |
Authors： |
TONG Jingcheng; |
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| Abstract： |
Let $\xi $ be an irrational number with simple continued fraction
expansion $$\xi =[a_0;a_1,\cdots,a_i,\cdots]$$ and
$\frac{p_i}{q_i}$ be its $i$th convergent. Let $C_i$ be defined by
$\xi -\frac{p_i}{q_i}=(-1)^i/(C_iq_iq_{i+1}).$ The author proves
the following theorem:
{\bf Theorem.} Let $r>1,R>1$ be two real numbers and\vskip 6pt
$L=\frac 1{r-1}+\frac 1{R-1}+a_na_{n+1}rR,\quad K=\frac
12\big(L+\sqrt{L^2-\frac 4{(r-1)(R-1)}}\;\big).$\vskip 6pt
\noindent Then
(\,i\,) $C_{n-2}K;$
(ii) $C_{n-2}>r,\ C_n>R$\ \ imply\ \ $C_{n-1} |
Keywords： |
Diophantine approximation, Simple continued fraction expansion |
Classification： |
11J70, 11K60 |
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