论文标题
在任意基础上由项链产生的正常数的差异界限
Discrepancy bounds for normal numbers generated by necklaces in arbitrary base
论文作者
论文摘要
Mordechay B. Levin构建了一个数字$λ$，这在基本2中是正常的，因此序列$（\ left \ {2^nλ\ right \}）_ {n = 0,1,1,2，\ ldots} $具有很小的差异$ d_n $。实际上，我们有$ n \ cdot d_n = \ mathcal {o} \ left（\ left（\ log n \ right）^2 \ right）$。莱文（Levin）的这种构建技术是由贝格（Becher）和卡顿（Carton）推广的，他们通过完美的嵌套项链产生了正常数字，他们表明，对于这些正常数字，相同的上部差异估计值与列文的特殊例子相同。在本文中，我们得出了一个上的上层差异，这些差异限制为所谓的半完美的嵌套项链，并表明，对于列文在任意的质量基本中的正常数量，对于差异的上限是最好的，即$ n \ cdot d_n \ cdot d_n \ geq c \ weft（\ cdot d_n \ geq c \ weft）该结果概括了先前的结果，我们确保为levin的特殊示例$ p = 2 $，$ n \ cdot d_n = o（\ left（\ log n \ right）^2）$在$ n $中是最好的。到目前为止，Schmidt的著名结果知道，对于$ [0,1）$，$ n \ cdot d_n \ geq c \ log n $的任何序列，带有$ c> 0 $，用于无限的许多$ n $。因此，在这个问题中存在$ \ log n $因素的差距，在$ n $中差异的最佳顺序是什么是正常数字。一方面，我们对列文在任何素数基础上的正常数字的结果可能支持猜测$ o（\ left（\ log n \ right）^2）$是$ n $中正常数字可以实现的最佳订单，同时通过引入已知正常数的类别来概括。另一方面，半完美的项链可能有助于搜索满足$ n $中较小差异界限的正常数字，而不是$ n \ cdot d_n = o（\ left（\ log n \ right）^2）$。
Mordechay B. Levin has constructed a number $λ$ which is normal in base 2, and such that the sequence $(\left\{2^n λ\right\})_{n=0,1,2,\ldots}$ has very small discrepancy $D_N$. Indeed we have $N\cdot D_N = \mathcal{O} \left(\left(\log N\right)^2\right)$. This construction technique of Levin was generalized by Becher and Carton, who generated normal numbers via perfect nested necklaces, and they showed that for these normal numbers the same upper discrepancy estimate holds as for the special example of Levin. In this paper now we derive an upper discrepancy bound for so-called semi-perfect nested necklaces and show that for the Levin's normal number in arbitrary prime base $p$ this upper bound for the discrepancy is best possible, i.e., $N\cdot D_N \geq c\left(\log N\right)^2$ with $c>0$ for infinitely many $N$. This result generalizes a previous result where we ensured for the special example of Levin for the base $p=2$, that $N\cdot D_N =O( \left(\log N\right)^2)$ is best possible in $N$. So far it is known by a celebrated result of Schmidt that for any sequence in $[0,1)$, $N\cdot D_N\geq c \log N$ with $c>0$ for infinitely many $N$. So there is a gap of a $\log N$ factor in the question, what is the best order for the discrepancy in $N$ that can be achieved for a normal number. Our result for Levin's normal number in any prime base on the one hand might support the guess that $O( \left(\log N\right)^2)$ is the best order in $N$ that can be achieved by a normal number, while generalizing the class of known normal numbers by introducing e.g. semi-perfect necklaces on the other hand might help for the search of normal numbers that satisfy smaller discrepancy bounds in $N$ than $N\cdot D_N=O( \left(\log N\right)^2)$.
