2024/07/18 by Verónica Becher, Becher, Verónica, Nicole Graus +1 · 1 citation
Mathematics · Physics and Astronomy · #Analytic Number Theory Research #Advanced Mathematical Identities #Quantum Mechanics and Applications
paper · pdf · doi:10.48550/arxiv.2407.13114
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.