## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 58, Number 2 (2017), 215-220.

### Normal Numbers and Limit Computable Cantor Series

Achilles Beros and Konstantinos Beros

#### Abstract

Given any oracle, $A$, we construct a basic sequence $Q$, computable in the jump of $A$, such that no $A$-computable real is $Q$-distribution-normal. A corollary to this is that there is a ${\Delta}_{n+1}^{0}$ basic sequence with respect to which no ${\Delta}_{n}^{0}$ real is distribution-normal. As a special case, there is a limit computable sequence relative to which no computable real is distribution-normal.

#### Article information

**Source**

Notre Dame J. Formal Logic Volume 58, Number 2 (2017), 215-220.

**Dates**

Received: 8 April 2014

Accepted: 21 November 2014

First available in Project Euclid: 22 March 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1490148081

**Digital Object Identifier**

doi:10.1215/00294527-2017-0004

**Subjects**

Primary: 03D28: Other Turing degree structures

Secondary: 03D80: Applications of computability and recursion theory

**Keywords**

computability theory recursion theory Turing degrees number theory normal numbers Cantor series expansions basic series

#### Citation

Beros, Achilles; Beros, Konstantinos. Normal Numbers and Limit Computable Cantor Series. Notre Dame J. Formal Logic 58 (2017), no. 2, 215--220. doi:10.1215/00294527-2017-0004. https://projecteuclid.org/euclid.ndjfl/1490148081