Weyl の一様分布定理と Benford の法則

Arnol'd (1989) には次のような問題が載っています:

Consider the first digits of the numbers $2 ^ n$: $1,2,4,8,1,3,6,1,2,5,1,2,4,\dots$. Does the digit $7$ appear in this sequence? Which digit appears more often, $7$ or $8$? How many times more often?

これは Benford の法則と呼ばれます. まずは Does the digit $7$ appear in this sequence? という問だけ解決しておきましょう. $2 ^ {46}=70368744177664$ です.

参考文献

Arnol’d, V. I. (1989). Mathematical methods of classical mechanics (Weinstein, A., Vogtmann, K., Trans.). Springer-Verlag New York. (Original work published 1974).