Bailey, D.H. and Borwein, Jonathan M. (2012) *Nonnormality of Stoneham constants.* Ramanujan J., 29 . pp. 409-422.

## Abstract

This paper examines “Stoneham constants,” namely real numbers of the form $α_{b,c} =\sum_{n\geq 1}1/(c^nb^{c^n})$, for coprime integers b ≥ 2 and c ≥ 2. These are of interest because, according to previous studies, αb,c is known to be bnormal, 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 $α_{2,3} =\sum_{n≥1} 1/(3^n2^{3^n})$ is 2-normal. More recently it was established that αb,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 $B = b^pc^qr$, 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 αb,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.

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