- Title
- Nonnormality of Stoneham constants
- Creator
- Bailey, David H.; Borwein, Jonathan M.
- Relation
- The Ramanujan Journal Vol. 29, p. 409-422
- Publisher Link
- http://dx.doi.org/10.1007/s11139-012-9417-3
- Publisher
- Springer New York
- Resource Type
- journal article
- Date
- 2012
- Description
- This paper examines “Stoneham constants,” namely real numbers of the form αb,c = Σn≥1 1/(cnbcn), for coprime integers b ≥ 2 and c ≥ 2. These are of interest because, according to previous studies, αb,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 α2,3 = Σn≥1 1/(3n23n) 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 = bpcq r, 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.
- Subject
- normality of irrational numbers; nonnormality of irrational numbers; Stoneham numbers; normality of sums
- Identifier
- http://hdl.handle.net/1959.13/940013
- Identifier
- uon:12923
- Identifier
- ISSN:1382-4090
- Rights
- The final publication is available at www.springerlink.com
- Language
- eng
- Full Text
- Reviewed
- Hits: 1328
- Visitors: 1662
- Downloads: 383
| Thumbnail | File | Description | Size | Format | |||
|---|---|---|---|---|---|---|---|
| View Details Download | ATTACHMENT02 | Author final version | 482 KB | Adobe Acrobat PDF | View Details Download |