Nonnormality of Stoneham constants

• Published in: The Ramanujan journal Vol. 29; no. 1-3; pp. 409 - 422
• Main Authors: Bailey, David H., Borwein, Jonathan M.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.12.2012
• Subjects: Combinatorics; Field Theory and Polynomials; Fourier Analysis; Functions of a Complex Variable; Mathematics; Mathematics and Statistics; Number Theory; 11K31; Normality of sums; 11K16; Normality of irrational numbers; Nonnormality of irrational numbers; Stoneham numbers
• ISSN: 1382-4090; 1572-9303
• DOI: 10.1007/s11139-012-9417-3

This paper examines “Stoneham constants,” namely real numbers of the form , for coprime integers b ≥2 and c ≥2. These are of interest because, according to previous studies, α b , c is known to be b -normal, meaning that every m -long string of base- b digits appears in the base- b expansion of the constant with precisely the limiting frequency b − m . So, for example, the constant is 2-normal. More recently it was established that α b , c is not bc -normal, so, for example, α 2,3 is provably not 6-normal. In this paper, we extend these findings by showing that α b , c is not B -normal, where B = b p c q r , for integers b and c as above, p , q , r ≥1, neither b nor c divide r , and the condition D = c q / p r 1/ p / b c −1 <1 is satisfied. It is not known whether or not this is a complete catalog of bases to which α b , c is nonnormal. We also show that the sum of two B -nonnormal Stoneham constants as defined above, subject to some restrictions, is B -nonnormal.