论文标题
广义rudin-shapiro序列的第2阶的离散相关性：一种组合方法
Discrete correlations of order 2 of generalised Rudin-Shapiro sequences: a combinatorial approach
论文作者
论文摘要
我们介绍了一个块状自动序列的家族，这些序列是通过向每个数字分配重量来获得的，并将序列的$ n $ th $ n $定义为以base $ k $写的整数$ n $的总重量。在权重函数的附加差异条件下，这些序列可以解释为概括性的rudin-shapiro序列，我们证明它们具有与符号序列相同的相关性，以均匀和独立的符号序列随机选择。收敛速度非常快，并且独立于$ k $的主要因素分解。这扩展了Tahay的最新工作。证明依赖于对整数和组合注意事项的基础$ K $表示的直接观察。我们还将结果扩展到更高的增长层序列。
We introduce a family of block-additive automatic sequences, that are obtained by allocating a weight to each couple of digits, and defining the $n$th term of the sequence as being the total weight of the integer $n$ written in base $k$. Under an additional difference condition on the weight function, these sequences can be interpreted as generalised Rudin-Shapiro sequences, and we prove that they have the same correlations of order 2 as sequences of symbols chosen uniformly and independently at random. The speed of convergence is very fast and is independent of the prime factor decomposition of $k$. This extends recent work of Tahay. The proof relies on direct observations about base-$k$ representations of integers and combinatorial considerations. We also provide extensions of our results to higher-dimensional block-additive sequences.
