# First digit phenomenon and ergodic theory

## Abstract

For any k belongs to I/sup +/ identical with the set of positive integers, let S(k) denote the set of positive integers beginning with k, i.e., S(k) = )n: k = (n/10/sup r/) for some r belongs to I/sup +/), where (x) denotes the greatest integer less than or equal to x. Consider some finitely additive extension of the density function d(S) = (lim/n ..-->.. infinity) (S intersection (1, N))/N, where S contains or equals I/sup +/ and (S) denotes the cardinality of S. If d(k) = d(S(k)) is defined for all positive integers K, and if the mapping T that maps S into (2S) union (2S + 1) preserves the density d, then the result of Cohen is that d(p) = log/sub 10/ (1 + (1/p)) for p belongs to D. The crucial step here is the specific relation of the transformation T to the multiplication of the elements of S by 2. This motivates us to consider the following problem: Set b = 10 and let a belongs to R/sup +/ be a positive real number. Let theta (x) = ax, where x belongs R/sup +/, and consider the sequence p = (p/sub 1/, p/sub 2/,...) of firstmore »

- Authors:

- Publication Date:

- Research Org.:
- Univ. of California, Santa Barbara

- OSTI Identifier:
- 5936327

- DOE Contract Number:
- W-7405-ENG-26

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: J. Math. Anal. Appl.; (United States); Journal Volume: 95:2

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ERGODIC HYPOTHESIS; ENTROPY; MATHEMATICS; RANDOMNESS; TRANSFORMATIONS; HYPOTHESIS; PHYSICAL PROPERTIES; THERMODYNAMIC PROPERTIES; 990200* - Mathematics & Computers

### Citation Formats

```
Robertson, J.B., Uppuluri, V.R.R., and Rajagopal, A.K.
```*First digit phenomenon and ergodic theory*. United States: N. p., 1983.
Web. doi:10.1016/0022-247X(83)90113-0.

```
Robertson, J.B., Uppuluri, V.R.R., & Rajagopal, A.K.
```*First digit phenomenon and ergodic theory*. United States. doi:10.1016/0022-247X(83)90113-0.

```
Robertson, J.B., Uppuluri, V.R.R., and Rajagopal, A.K. Thu .
"First digit phenomenon and ergodic theory". United States.
doi:10.1016/0022-247X(83)90113-0.
```

```
@article{osti_5936327,
```

title = {First digit phenomenon and ergodic theory},

author = {Robertson, J.B. and Uppuluri, V.R.R. and Rajagopal, A.K.},

abstractNote = {For any k belongs to I/sup +/ identical with the set of positive integers, let S(k) denote the set of positive integers beginning with k, i.e., S(k) = )n: k = (n/10/sup r/) for some r belongs to I/sup +/), where (x) denotes the greatest integer less than or equal to x. Consider some finitely additive extension of the density function d(S) = (lim/n ..-->.. infinity) (S intersection (1, N))/N, where S contains or equals I/sup +/ and (S) denotes the cardinality of S. If d(k) = d(S(k)) is defined for all positive integers K, and if the mapping T that maps S into (2S) union (2S + 1) preserves the density d, then the result of Cohen is that d(p) = log/sub 10/ (1 + (1/p)) for p belongs to D. The crucial step here is the specific relation of the transformation T to the multiplication of the elements of S by 2. This motivates us to consider the following problem: Set b = 10 and let a belongs to R/sup +/ be a positive real number. Let theta (x) = ax, where x belongs R/sup +/, and consider the sequence p = (p/sub 1/, p/sub 2/,...) of first digits of the orbit of theta: )x, theta(x), theta/sup 2/ (x), ...). That is p/sub k/ = (theta /sup k/ (x)/b/sup r/) belongs to D where r = (log/sub 10/ theta/sup k/ (x)). The purpose of this paper is to investigate the dynamical properties of the sequence p. 7 references.},

doi = {10.1016/0022-247X(83)90113-0},

journal = {J. Math. Anal. Appl.; (United States)},

number = ,

volume = 95:2,

place = {United States},

year = {Thu Sep 01 00:00:00 EDT 1983},

month = {Thu Sep 01 00:00:00 EDT 1983}

}