Nonnormality of Stoneham constants - Bailey, David H.; Borwein, Jonathan M. (The Ramanujan journal, 2012)
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.