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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2506.15422 |
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| _version_ | 1866911615635423232 |
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| author | Manai, Chokri |
| author_facet | Manai, Chokri |
| contents | We introduce and study a new topological notion of the size for subsets of the real line, called \emph{super-density}. A set $A\subset\mathbb{R}$ is super-dense if for every non-empty open interval $I$ and every nowhere constant continuous function $φ\colon I\to\mathbb{R}$, we have $φ(I\cap A)\cap A\neq\emptyset$. We first establish basic properties of super-dense sets. Our main topological result characterizes them within the framework of Baire category: a set with the Baire property is super-dense if and only if it is co-meager. We then investigate the implications for the theory of normal numbers. We prove that the set of non-normal numbers is super-dense, whereas the set of normal numbers is not. Consequently, no nowhere constant continuous function can map all non-normal numbers to normal numbers. Conversely, we explicitly construct a computable nowhere constant continuous function that maps all normal numbers to non-normal numbers. Finally, we provide a constructive algorithm that, given any countable family of nowhere constant continuous functions, produces a real number $x$ such that $x$ and all its images under these functions are non-normal. As a corollary, we obtain the existence of a non-normal number $x$ such that $e^{αx}$ is non-normal for every non-zero algebraic $α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15422 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Super-Dense Sets and Their Role in the Theory of Normal Numbers Manai, Chokri Number Theory General Topology 11K16, 26A30 We introduce and study a new topological notion of the size for subsets of the real line, called \emph{super-density}. A set $A\subset\mathbb{R}$ is super-dense if for every non-empty open interval $I$ and every nowhere constant continuous function $φ\colon I\to\mathbb{R}$, we have $φ(I\cap A)\cap A\neq\emptyset$. We first establish basic properties of super-dense sets. Our main topological result characterizes them within the framework of Baire category: a set with the Baire property is super-dense if and only if it is co-meager. We then investigate the implications for the theory of normal numbers. We prove that the set of non-normal numbers is super-dense, whereas the set of normal numbers is not. Consequently, no nowhere constant continuous function can map all non-normal numbers to normal numbers. Conversely, we explicitly construct a computable nowhere constant continuous function that maps all normal numbers to non-normal numbers. Finally, we provide a constructive algorithm that, given any countable family of nowhere constant continuous functions, produces a real number $x$ such that $x$ and all its images under these functions are non-normal. As a corollary, we obtain the existence of a non-normal number $x$ such that $e^{αx}$ is non-normal for every non-zero algebraic $α$. |
| title | Super-Dense Sets and Their Role in the Theory of Normal Numbers |
| topic | Number Theory General Topology 11K16, 26A30 |
| url | https://arxiv.org/abs/2506.15422 |