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On Normality Preserving Operations

  • Chokri Manai
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Abstract

We study arithmetic operations and maps which preserve normality. Revisiting a recent argument of Dayan, Ganguly and Weiss, we give a streamlined, self-contained presentation of the probabilistic proof that the non-zero rationals are exactly the multiplicative normality preserving numbers in a fixed integer base. We also include the extension to all bases simultaneously, using the theorem of Hochman and Shmerkin. In the additive case, we extend Rauzy's characterization of deterministic numbers in a fixed integer base to absolutely normal numbers. Namely, a translation preserves absolute normality if and only if its parameter is deterministic in every integer base. We prove this extension by a sparse random perturbation argument. Finally, we show that all locally C2 normality preserving maps need to be affine-linear. The regularity assumption cannot be weakened to C1,1: we construct a non-affine C1,1 diffeomorphism which preserves normality in every integer base, and hence absolute normality.

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Cite

BibTeXRIS
BibTeX
@misc{manai2026normality,
  title = {{On Normality Preserving Operations}},
  author = {Chokri Manai},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24665},
  eprint = {2609.24665},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24665},
}
RIS
TY  - UNPB
TI  - On Normality Preserving Operations
AU  - Chokri Manai
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24665
UR  - https://arxiv.org/abs/2609.24665
AB  - We study arithmetic operations and maps which preserve normality. Revisiting a recent argument of Dayan, Ganguly and Weiss, we give a streamlined, self-contained presentation of the probabilistic proof that the non-zero rationals are exactly the multiplicative normality preserving numbers in a fixed integer base. We also include the extension to all bases simultaneously, using the theorem of Hochman and Shmerkin. In the additive case, we extend Rauzy's characterization of deterministic numbers in a fixed integer base to absolutely normal numbers. Namely, a translation preserves absolute normality if and only if its parameter is deterministic in every integer base. We prove this extension by a sparse random perturbation argument. Finally, we show that all locally C^2 normality preserving maps need to be affine-linear. The regularity assumption cannot be weakened to C^1,1: we construct a non-affine C^1,1 diffeomorphism which preserves normality in every integer base, and hence absolute normality.
ER  -