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On the discrepancy and pseudorandomness of new sequences. (Chinese. English summary) Zbl 1174.11063

Summary: Let \(p\) be an odd prime, and \(x\) be an integer with \(1\leq x\leq p-1\). Define \[ x_n = \begin{cases} \{\frac{\overline{n}+\overline{n+x}}{p}\},& \text{if} \quad p \nmid n(n+x),\\ 0, & \text{otherwise} \end{cases} \] and \[ e_n=\begin{cases} +1, \quad\text{if} \quad p \nmid n(n+x)\,\quad\text{and}\quad 0\leqslant \{\bar{n}+\overline{n+x} \} < \frac{1}{2}, \\-1, \quad\text{if}\quad p \nmid n(n+x) \quad\text{and}\quad\frac12\leqslant \{\bar{n}+\overline{n+x}\}<1, \\ +1 , \quad\text{if}\quad p\,\,|\,n(n+x), \end{cases} \] where \(\bar{n}\) is the multiplicative inverse of \(n\) modulo \(p\) such that \(1\leq n \leq p-1\) . This paper proves that \((x_n)\) is uniformly distributed modulo \(1\), and \((e_n)\) is a “good” pseudorandom sequence. This shows that there are some links between finite binary sequences and \([0, 1)\) sequences.

MSC:

11K38 Irregularities of distribution, discrepancy
11K45 Pseudo-random numbers; Monte Carlo methods
11K31 Special sequences
11A07 Congruences; primitive roots; residue systems
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