We generalize the construction of Roy’s Fibonacci type numbers to the case of a Sturmian recurrence and we determine the classical exponents of approximation $$\omega _2(\xi )$$ ω 2 ( ξ ) , $${\widehat{\omega }}_2(\xi )$$ ω ^ 2 ( ξ ) , $$\lambda _2(\xi )$$ λ 2 ( ξ ) , $${\widehat{\lambda }}_2(\xi )$$ λ ^ 2 ( ξ ) associated with these real numbers. This also extends similar results established by Bugeaud and Laurent in the case of Sturmian continued fractions. More generally we provide an almost complete description of the combined graph of parametric successive minima functions defined by Schmidt and Summerer in dimension two for such Sturmian type numbers. As a side result we obtain new information on the joint spectra of the above exponents as well as a new family of numbers for which it is possible to construct the sequence of the best rational approximations.

中文翻译：

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Sturmian 类型数在维度 2 中的丢番图近似指数

我们将 Roy 的斐波那契类型数的构造推广到 Sturmian 递归的情况，并确定了近似值的经典指数 $$\omega _2(\xi )$$ ω 2 ( ξ ) , $${\widehat{\omega } }_2(\xi )$$ ω ^ 2 ( ξ ) , $$\lambda _2(\xi )$$ λ 2 ( ξ ) , $${\widehat{\lambda }}_2(\xi )$$ λ ^ 2 ( ξ ) 与这些实数相关。这也扩展了 Bugeaud 和 Laurent 在 Sturmian 连分数的情况下建立的类似结果。更一般地说，我们几乎完整地描述了 Schmidt 和 Summerer 为此类 Sturmian 类型数在第二维中定义的参数连续最小值函数的组合图。