Nonnormality of Stoneham constants

Abstract

This paper examines “Stoneham constants,” namely real numbers of the form \(\alpha_{b,c} = \sum_{n \geq1} 1/(c^{n} b^{c^{n}})\), 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 \(\alpha_{2,3} = \sum_{n \geq1} 1/(3^{n} 2^{3^{n}})\) 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.