Markoff-Lagrange spectrum uncovers exotic topological properties of Diophantine approximation. We investigate asymptotic properties of geometric progressions modulo one and observe significantly analogous results on the set$\mathcal{L}(\alpha )=\{\underset{n\to \infty }{\mathrm{lim\hspace{0.17em}sup}}\Vert \xi {\alpha }^n\Vert |\xi \in \mathbb{R}\},$ where $\Vert x\Vert$ is the distance from x to the nearest integer. First, we show that $\mathcal{L}(\alpha )$ is closed in $[0,1/2]$ for any Pisot number α.

Then we consider the case where α is an integer with $\alpha \ge 2$, or a quadratic unit with $\alpha \ge 3$. We show that $\mathcal{L}(\alpha )$ contains a proper interval when α is quadratic but it does not when α is an integer. We also determine the minimum limit point and all isolated points beneath this point. In the course of the proof, we revisit a property studied by Markoff which characterizes bi-infinite balanced words and sturmian words.

Markoff-Lagrange光谱揭示了Diophantine逼近的奇异拓扑特性。我们研究模数为1的几何级数的渐近性质，并在集合上观察到明显相似的结果$\mathcal{大号}（\alpha ）=\{\underset{ñ\to \infty }{\mathrm{lim\; sup}}\Vert \xi {\alpha }^{ñ}\Vert |\xi \in \mathbb{[R}\}，$ 哪里 $\Vert X\Vert$是从x到最接近的整数的距离。首先，我们证明$\mathcal{大号}（\alpha ）$ 在关闭 $[0，\mathrm{1个}/2]$对于任何Pisot数α。

然后我们考虑α是一个整数$\alpha \ge 2$，或具有 $\alpha \ge 3$。我们证明$\mathcal{大号}（\alpha ）$当α是二次数时，包含一个适当的间隔，但是当α是整数时，它不包含一个适当的间隔。我们还确定最小极限点和该点以下的所有孤立点。在证明过程中，我们将重新审视Markoff研究的属性，该属性表征双无限平衡单词和turmian单词。