Interleave sequence

In mathematics, an interleave sequence is obtained by merging or shuffling two sequences.

Let $S$ be a set, and let $(x_i)$ and $(y_i)$, $i=0,1,2,\ldots,$ be two sequences in $S.$ The interleave sequence is defined to be the sequence $x_0, y_0, x_1, y_1, \dots.$ Formally, it is the sequence $(z_i), i=0,1,2,\ldots$ given by

$z_i := \begin{cases} x_k & \text{ if } i=2k \text{ is even,}\\ y_k & \text{ if } i=2k+1 \text{ is odd.} \end{cases}$

Properties

• The interleave sequence $(z_i)$ is convergent if and only if the sequences $(x_i)$ and $(y_i)$ are convergent and have the same limit.^[1]
• Consider two real numbers a and b greater than zero and smaller than 1. One can interleave the sequences of digits of a and b, which will determine a third number c, also greater than zero and smaller than 1. In this way one obtains an injection from the square (0, 1)×(0, 1) to the interval (0, 1). Different radixes give rise to different injections; the one for the binary numbers is called the Z-order curve or Morton code.^[2]

References

1. Strichartz, Robert S. (2000), The Way of Analysis, Jones & Bartlett Learning, p. 78, ISBN 9780763714970.
2. Mamoulis, Nikos (2012), Spatial Data Management, Synthesis lectures on data management 21, Morgan & Claypool Publishers, pp. 22–23, ISBN 9781608458325.

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