### Mori, Y and Takashima, K (2016)

#### On the distribution of the leading digit of a^{n}: a study via 𝜒^{2} statistics

Period. Math. Hungar. 73(2), 224-239.

**ISSN/ISBN:** 0031-5303
**DOI:** 10.1007/s10998-016-0138-z

**Abstract:** We investigate irrational rotations with isolated large partial quotients from the point of view of the distribution of the leading digit of a^{n}. We prove some mathematical formulae explaining the unusual behavior of the 𝜒^{2} statistic of the leading digits of a^{n}, where log_{10} a has a single isolated large partial quotient in its continued fraction expansion. We also report that hills appear infinitely often in the graphs of 𝜒^{2}statistics and that there are many different types of shapes of hills.

**Bibtex:**

```
AUTHOR = {Mori, Yoshiyuki and Takashima, Keizo},
TITLE = {On the distribution of the leading digit of {$a^n$}: a study via {$\chi^2$} statistics},
JOURNAL = {Period. Math. Hungar.},
FJOURNAL = {Periodica Mathematica Hungarica. Journal of the J\'anos Bolyai Mathematical Society},
VOLUME = {73},
YEAR = {2016},
NUMBER = {2},
PAGES = {224--239},
ISSN = {0031-5303},
MRCLASS = {11K38 (11A55 11K31)},
MRNUMBER = {3564566},
MRREVIEWER = {Vilius Stak\.enas},
DOI = {10.1007/s10998-016-0138-z},
URL = {http://dx.doi.org/10.1007/s10998-016-0138-z},
}
```

**Reference Type:** Journal Article

**Subject Area(s):** Analysis, Number Theory, Statistics