On the distribution of the leading digit of $$a^n$$ a n : a study via $$\chi ^2$$ χ 2 statistics

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 $$\chi ^2$$ 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 $$\chi ^2$$ statistics and that there are many different types of shapes of hills.