Nonnormality of Stoneham constants

This paper examines “Stoneham constants,” namely real numbers of the form α_b,c = Σ_n≥1 1/(cⁿb^cn), 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 α_2,3 = Σn≥1 1/(3ⁿ2³ⁿ) 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^pc^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/pr^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.