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Multiplicative analogue of Markoff-Lagrange spectrum and Pisot numbers
Advances in Mathematics ( IF 1.5 ) Pub Date : 2021-01-13 , DOI: 10.1016/j.aim.2020.107547
Shigeki Akiyama , Hajime Kaneko

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 setL(α)={lim supnξαn|ξR}, where x is the distance from x to the nearest integer. First, we show that L(α) is closed in [0,1/2] for any Pisot number α.

Then we consider the case where α is an integer with α2, or a quadratic unit with α3. We show that L(α) 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.



中文翻译:

Markoff-Lagrange谱和Pisot数的乘法模拟

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大号α={lim supñξαñ|ξ[R} 哪里 X是从x到最接近的整数的距离。首先,我们证明大号α 在关闭 [01个/2]对于任何Pisot数α

然后我们考虑α是一个整数α2,或具有 α3。我们证明大号αα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。

更新日期:2021-01-13
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