Nonnormality of Stoneham constants

作者：Bailey David H; Borwein Jonathan M^*

来源：Ramanujan Journal, 2012, 29(1-3): 409-422.

DOI：10.1007/s11139-012-9417-3

• Normality of irrational numbers
• Nonnormality of irrational numbers
• Stoneham numbers
• Normality of sums

This paper examines %26quot;Stoneham constants,%26quot; namely real numbers of the form , for coprime integers ba parts per thousand yen2 and ca parts per thousand yen2. These are of interest because, according to previous studies, alpha (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 alpha (b,c) is not bc-normal, so, for example, alpha (2,3) is provably not 6-normal. In this paper, we extend these findings by showing that alpha (b,c) is not B-normal, where B=b (p) c (q) r, for integers b and c as above, p,q,ra parts per thousand yen1, neither b nor c divide r, and the condition D=c (q/p) r (1/p) /b (c-1)%26lt; 1 is satisfied. It is not known whether or not this is a complete catalog of bases to which alpha (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.

• 出版日期2012-12