Arithmetical properties of the digits of the multiples of an irrational number

Abstract

Little seems to be known about the digits or sequences of digits in the decimal representation of a given irrational number like √2 or π. There is no difficulty in constructing an irrational number such that in its decimal representation certain digits or sequences of digits do not occur. On the other hand, well known theorems by Tchebychef, Kronecker, and Weyl imply that some integral multiple of the given irrational number always has any given finite sequence of digits occuring at least once in its decimal representation: for the fractional parts of the multiplies of the number lie dense in the interval (0, 1).