正则Pisot数的β展开
Beta Expansions for Regular Pisot Numbers
AI总结：
本文针对小于2的正则Pisot数，完整列出了1的贪婪β展开，揭示了显著模式，并解决了Boyd关于循环因子的猜想。
摘要：
β展开是实数基数为非整数的十进制表示的类比。虽然使用非整数基数的1的贪婪β展开通常无限长且不重复，但已知如果基数是Pisot数，则该展开总是有限或周期的。已有一些工作致力于了解这些展开，但通常这些展开并不明确已知。在本文中，我们给出了基数为任何小于2的正则Pisot数时1的贪婪β展开的完整列表，揭示了各种显著模式。我们还回答了Boyd关于贪婪展开的循环因子的猜想。
英文摘要：
A beta expansion is the analogue of the base 10 representation of a real number, where the base may be a non-integer. Although the greedy beta expansion of 1 using a non-integer base is in general infinitely long and non-repeating, it is known that if the base is a Pisot number, then this expansion will always be finite or periodic. Some work has been done to learn more about these expansions, but in general these expansions were not explicitly known. In this paper, we present a complete list of the greedy beta expansions of 1 where the base is any regular Pisot number less than 2, revealing a variety of remarkable patterns. We also answer a conjecture of Boyd's regarding cyclotomic co-factors for greedy expansions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
