On the distance from a rational power to the nearest integer
On the distance from a rational power to the nearest integer
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关于有理数幂到最接近整数的距离
DOI：
10.1016/j.jnt.2005.07.004
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发表时间：
2006
影响因子：
0.7
通讯作者：
A. Dubickas
中科院分区：
文献类型：
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作者：
A. Dubickas
证明了对任意非零真实的数n，小数部分序列{n（3/2）n}，n= 1，2，3，.，在长度为0.523764.的区间[0.238117.中至少包含一个极限点.更一般地说，它表明，每个序列的距离最近的整数||n（p/q）n||，n= 1，2，3，...，其中p/q>1为有理数，有“大”极限点和“小”极限点。所有得到的常数都用p和q明确表示。它们也可以用Thue-Morse序列表示，并且对于无理数，对于每对p>1，q=1，它们都是最好的可能。此外，我们对任意序列给出了类似的有效结果，从而加强了Pisot和Vijayaraghavan的一个经典结果||αn||，n= 1，2，3，.，其中α>1是一个代数数，而当α是一个Pisot数或Salem数时，α 0是一个任意的真实的数，它满足α = Q（α）.
We prove that for any non-zero real number ξ the sequence of fractional parts {ξ(3/2)n}, n=1,2,3,…, contains at least one limit point in the interval [0.238117…,0.761882…] of length 0.523764…. More generally, it is shown that every sequence of distances to the nearest integer ||ξ(p/q)n||, n=1,2,3,…, where p/q>1 is a rational number, has both ‘large’ and ‘small’ limit points. All obtained constants are explicitly expressed in terms of p and q. They are also expressible in terms of the Thue–Morse sequence and, for irrational ξ, are best possible for every pair p>1, q=1. Furthermore, we strengthen a classical result of Pisot and Vijayaraghavan by giving similar effective results for any sequence ||ξαn||, n=1,2,3,…, where α>1 is an algebraic number and where ξ≠0 is an arbitrary real number satisfying ξ∉Q(α) in case α is a Pisot or a Salem number.