Nonnormality of Stoneham constants

This paper examines “Stoneham constants,” namely real numbers of the form αb,c = ∑ n≥1 1/(c nbc 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 α2,3 = ∑ n≥1 1/(3 n23 n ) is 2-normal. More recently it was established that αb,c is not bc-normal, so, for example, α2,3 is provably not 6-normal. In this paper, we extend these findings by showing that αb,c is not B-normal, where B = bpcqr, for integers b and c as above, p, q, r ≥ 1, neither b nor c divide r, and the condition D = cq/pr1/p/bc−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.