Powers of rationals modulo 1 and rational base number systems
Powers of rationals modulo 1 and rational base number systems
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DOI：
10.1007/s11856-008-1056-4
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发表时间：
2008-01-01
影响因子：
1
通讯作者：
Sakarovitch, Jacques
中科院分区：
文献类型：
--
作者：
Akiyama, Shigeki;Frougny, Christiane;Sakarovitch, Jacques
考虑一种在有理基中表示正整数和实数的新方法。它相当于从右到左计算数字，首先是最不重要的。每个整数都有唯一的扩展。整数的扩展集不是常规语言，但是加法可以通过字母到字母的有限右变换器来执行。每个实数都至少有一个这样的展开式，并且可数无限个实数有多个这样的展开式。我们解释了如何近似这些展开式，并描述具有两个展开式的实数展开式。我们得出的结果本身是相关的，而且还与组合学和数论中的其他问题相关。第一个例子是对所谓“约瑟夫问题”中的常数 K(p) 的新解释和扩展。更重要的是，p/q 基数的这些扩展使我们能够在有理数幂小数部分的分布问题上取得一些进展。
A new method for representing positive integers and real numbers in a rational base is considered. It amounts to computing the digits from right to left, least significant first. Every integer has a unique expansion. The set of expansions of the integers is not a regular language but nevertheless addition can be performed by a letter-to-letter finite right transducer. Every real number has at least one such expansion and a countable infinite number of them have more than one. We explain how these expansions can be approximated and characterize the expansions of reals that have two expansions.The results that we derive are pertinent on their own and also as they relate to other problems in combinatorics and number theory. A first example is a new interpretation and expansion of the constant K(p) from the so-called "Josephus problem." More important, these expansions in the base p/q allow us to make some progress in the problem of the distribution of the fractional part of the powers of rational numbers.