### Abstract

Let T and S be two number theoretical transformations on the n-dimensional unit cube B, and write T∼S if there exist positive integers m and n such that T^{m}=S^{n}. F. Schweiger showed in [1969, J. Number Theory1, 390-397] that T∼S implies that every T-normal number x is S-normal. Furthermore, he conjectured that T≁S implies that not all T-normal x are S-normal. In this note two counterexamples to this conjecture are given.

Original language | English |
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Pages (from-to) | 330-340 |

Number of pages | 11 |

Journal | Journal of Number Theory |

Volume | 86 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2001 Feb |

### Fingerprint

### Keywords

- Normal numbers

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

*86*(2), 330-340. https://doi.org/10.1006/jnth.2000.2554

**On a problem of Schweiger concerning normal numbers.** / Kraaikamp, Cor; Nakada, Hitoshi.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 86, no. 2, pp. 330-340. https://doi.org/10.1006/jnth.2000.2554

}

TY - JOUR

T1 - On a problem of Schweiger concerning normal numbers

AU - Kraaikamp, Cor

AU - Nakada, Hitoshi

PY - 2001/2

Y1 - 2001/2

N2 - Let T and S be two number theoretical transformations on the n-dimensional unit cube B, and write T∼S if there exist positive integers m and n such that Tm=Sn. F. Schweiger showed in [1969, J. Number Theory1, 390-397] that T∼S implies that every T-normal number x is S-normal. Furthermore, he conjectured that T≁S implies that not all T-normal x are S-normal. In this note two counterexamples to this conjecture are given.

AB - Let T and S be two number theoretical transformations on the n-dimensional unit cube B, and write T∼S if there exist positive integers m and n such that Tm=Sn. F. Schweiger showed in [1969, J. Number Theory1, 390-397] that T∼S implies that every T-normal number x is S-normal. Furthermore, he conjectured that T≁S implies that not all T-normal x are S-normal. In this note two counterexamples to this conjecture are given.

KW - Normal numbers

UR - http://www.scopus.com/inward/record.url?scp=0035255498&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035255498&partnerID=8YFLogxK

U2 - 10.1006/jnth.2000.2554

DO - 10.1006/jnth.2000.2554

M3 - Article

AN - SCOPUS:0035255498

VL - 86

SP - 330

EP - 340

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 2

ER -