Values of generalized Liouville power series at algebraic numbers
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912487346012160 |
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| author | Bilu, Yu. Marques, D. Moreira, C. G. |
| author_facet | Bilu, Yu. Marques, D. Moreira, C. G. |
| contents | For every positive integer $m$ LeVeque (1953) defined the $U_m$-numbers as the transcendental numbers that admit very good approximation by algebraic numbers of degree $m$, but not by those of smaller degree. In these terms, Mahler's $U$-numbers are the transcendental numbers which are $U_m$ for some $m$. In 1965 Mahler showed that (properly defined) lacunary power series with integers coefficients take $U$-values at algebraic numbers, unless the value is algebraic for an obvious reason. However, his argument does not specify to which $U_m$ the value belongs.
In this article, we introduce the notion of generalized Liouville series, and give a necessary and sufficient condition for their values to be $U_m$. As an application, we show that a generalized Liouville series takes a $U_m$-value at a simple algebraic integer of degree $m$, unless the value is algebraic for an obvious reason. (An algebraic number $α$ is called simple if the number field $\mathbb Q(α)$ does not have a proper subfield other than $\mathbb Q$.) |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_20530 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Values of generalized Liouville power series at algebraic numbers Bilu, Yu. Marques, D. Moreira, C. G. Number Theory For every positive integer $m$ LeVeque (1953) defined the $U_m$-numbers as the transcendental numbers that admit very good approximation by algebraic numbers of degree $m$, but not by those of smaller degree. In these terms, Mahler's $U$-numbers are the transcendental numbers which are $U_m$ for some $m$. In 1965 Mahler showed that (properly defined) lacunary power series with integers coefficients take $U$-values at algebraic numbers, unless the value is algebraic for an obvious reason. However, his argument does not specify to which $U_m$ the value belongs. In this article, we introduce the notion of generalized Liouville series, and give a necessary and sufficient condition for their values to be $U_m$. As an application, we show that a generalized Liouville series takes a $U_m$-value at a simple algebraic integer of degree $m$, unless the value is algebraic for an obvious reason. (An algebraic number $α$ is called simple if the number field $\mathbb Q(α)$ does not have a proper subfield other than $\mathbb Q$.) |
| title | Values of generalized Liouville power series at algebraic numbers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2505.20530 |