Markoff-Lagrange谱与极值数
Markoff-Lagrange spectrum and extremal numbers
摘要：
让伽玛表示黄金分割率。H. Davenport和W. M.Schmidt在1969年证明，对于每个非二次无理实数xi，存在一个常数c>0，其性质是，对于任意大的X值，不等式|x_0| \le X，|x_0*xi - x_1| \le cX^{-1/γ}，|x_0*xi^2 - x_2| \le cX^{-1/gamma}不允许非零整数解（x_0，x_1，x_2）。他们的结果是最好的可能的意义，相反，有可数许多非二次无理实数xi，使得对于较大的c值，相同的不等式允许每个X \ge 1的非零整数解。在GL_2（Z）作用下，通过线性分式变换，这类极值数是超越的，其集合是稳定的。本文证明了存在Lagrange常数为1/3的极值数xi，这是一个非二次数的最大可能值，并且证明了这类数的GL_2（Z）-等价类与Markoff方程的非平凡解之间存在一个自然的双射.
英文摘要：
Let gamma denote the golden ratio. H. Davenport and W. M.Schmidt showed in 1969 that, for each non-quadratic irrational real number xi, there exists a constant c>0 with the property that, for arbitrarily large values of X, the inequalities |x_0| \le X, |x_0*xi - x_1| \le cX^{-1/gamma}, |x_0*xi^2 - x_2| \le cX^{-1/gamma} admit no non-zero integer solution (x_0,x_1,x_2). Their result is best possible in the sense that, conversely, there are countably many non-quadratic irrational real numbers xi such that, for a larger value of c, the same inequalities admit a non-zero integer solution for each X\ge 1. Such extremal numbers are transcendental and their set is stable under the action of GL_2(Z) by linear fractional transformations. In this paper, it is shown that there exists extremal numbers xi for which the Lagrange constant is 1/3, the largest possible value for a non-quadratic number, and that there is a natural bijection between the GL_2(Z)-equivalence classes of such numbers and the non-trivial solutions of Markoff's equation.
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