Authors: V. Becher; N. Graus.
Abstract:
A number is normal in base b if, in its base b expansion, all blocks of digits of equal length have the same asymptotic frequency. The rate at which a number approaches normality is quantified by the classical notion of discrepancy, which indicates how far the scaling of the number by powers of b is from being equidistributed modulo 1.
This rate is known as the discrepancy of a normal number. The Champernowne constant C10 = 0.12345678910111213141516. . . is the most well-known example of a normal number. In 1986, Schiffer provided the discrepancy of numbers in a family that includes the
Champernowne constant. His proof relies on exponential sums. Here, we present a discrete and elementary proof specifically for the discrepancy of the Champernowne constant.
More information:
https://www-2.dc.uba.ar/staff/becher/papers/champ.pdf
