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# Analogues of the van der Corput’s sequence

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van der Corput sequence

A van der Corput sequence is a low-discrepancy sequence over the unit interval first published in 1935 by the Dutch mathematician J. G. van der Corput. It is constructed by reversing the base $b$ representation of the sequence of natural numbers. For example, the decimal van der Corput sequence begins:

$0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.01, 0.11, \ldots$

van der Corput proved the following theorem:

Theorem 1: The sequence $v(n)$ is equidistributed modulo one.

As a corollary of the above theorem, it follows from the property of sequence equidistributed modulo one that the mean value of $v(n)$ approaches 0.5 as n tends to infinity i.e.

$\displaystyle{\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{r=1}^{n}v(r) = \frac{1}{2}.}$

Generalized van der Corput sequence

In the same spirit, we can form van der Corput like sequence like by reversing the base $b$ representation of the sequence of primes or the sequence of composite numbers or any other sequence and study the properties of these analogues of the van der Corput sequence. We define a generalization of the van der Corput type sequences as follows:

Definition: Let $a_n$ be the base $b$ representation of a sequence of positive reals defined for $n \ge 1$. We define $v(a_n)$ as the sequence of numbers formed by placing a decimal point, and writing all the digits in the base $b$ representation of the integer part of $a_n$ in the reverse order after the decimal point.

Example: $v(p_n)$ is the sequence

$0.2, 0.3, 0.5, 0.7, 0.11, 0.31, 0.71, 0.91, 0.32, 0.92, \ldots$.

Since primes other than 2 do not end in an even number in the unit’s place so clearly the sequence $v(p_n)$ is clearly not equidistributed mod 1. In this case what will the limiting value of the mean value of $v(p_n)$? In general we want to know how for a give $a_n$, the sequence $v(a_n)$ will behave. Before we proceed with this, we will first need to little bit about normal numbers.

A number is said to be simply normal to base $b$ if its base $b$ expansion has each digit appearing with average frequency tending to $b^{-1}.$

Champernowne’s number $0.123456789101112...$

obtained by concatenating the decimal representations of the natural numbers in order, is normal in base 10, but it might not be normal in some other bases.

Definition: Let $a_n$ be a sequence of natural number in $b$. We define by $C_b (a_n)$ as the number formed by placing a decimal point and the concatenating the digits of the $b$ representation of $a_n$.

Example: For the sequence of prime numbers $p_n$,

$C_{10} (p_n) = 0.23571113171923...$,

which is also known as the Copeland–Erdős constant.

In this regard, we have the following result.

Theorem 2. If $a_n$ is a sequence of natural numbers in base 10 such that

1. the unit’s digit of $a_n$ can take any value form the set $D=\{d_1, d_2, ... , d_k\}$ where each $d_i \in \{0, 1, 2, ... , 9\}$.
2. $C_{10} (a_n)$ is normal in base 10

then

$\displaystyle{\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{r=1}^{n}v(a_r) = \frac{1}{20} + \frac{1}{10k}\sum_{i=1}^{k}d_i}.$

Corollary 1. Since the Copeland–Erdős constant $C_{10} (p_n)$ is normal in base 10 and all primes greater than 5 have 1, 3, 7 or 9 in the unit’s place,

$\displaystyle{\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{r=1}^{n}v(p_r)=\frac{11}{20}.}$

Corollary 2.

$\displaystyle{\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{r=1}^{n}v(p_r)=\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{r=1}^{n}v(2r+1)=\frac{11}{20}}.$

The above theorem is a special case of a more general result on partially equidistributed sequences about which I shal write in my next post. Interested readers can refer to my paper ‘Contributions to equidistribution theory.’