The $t$-metric Mahler measures of surds and rational numbers
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arXiv
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| Format: | Preprint |
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2014
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| _version_ | 1866913769453518848 |
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| author | Samuels, Charles L. Jankauskas, Jonas |
| author_facet | Samuels, Charles L. Jankauskas, Jonas |
| contents | A. Dubickas and C. Smyth introduced the metric Mahler measure $$
M_1(α) = \inf\left\{\sum_{n=1}^N M(α_n): N \in \mathbb N, α_1 \cdots α_N = α\right\}, $$ where $M(α)$ denotes the usual (logarithmic) Mahler measure of $α\in \overline{\mathbb Q}$. This definition extends in a natural way to the $t$-metric Mahler measure by replacing the sum with the usual $L_t$ norm of the vector $(M(α_1), \dots, M(α_N))$ for any $t\geq 1$. For $α\in \mathbb Q$, we prove that the infimum in $M_t(α)$ may be attained using only rational points, establishing an earlier conjecture of the second author. We show that the natural analogue of this result fails for general $α\in\overline{\mathbb Q}$ by giving an infinite family of quadratic counterexamples. As part of this construction, we provide an explicit formula to compute $M_t(D^{1/k})$ for a square-free $D \in \mathbb N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1408_4166 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | The $t$-metric Mahler measures of surds and rational numbers Samuels, Charles L. Jankauskas, Jonas Number Theory 11R04, 11R09 (Primary), 11C08, 12E05 (Secondary) A. Dubickas and C. Smyth introduced the metric Mahler measure $$ M_1(α) = \inf\left\{\sum_{n=1}^N M(α_n): N \in \mathbb N, α_1 \cdots α_N = α\right\}, $$ where $M(α)$ denotes the usual (logarithmic) Mahler measure of $α\in \overline{\mathbb Q}$. This definition extends in a natural way to the $t$-metric Mahler measure by replacing the sum with the usual $L_t$ norm of the vector $(M(α_1), \dots, M(α_N))$ for any $t\geq 1$. For $α\in \mathbb Q$, we prove that the infimum in $M_t(α)$ may be attained using only rational points, establishing an earlier conjecture of the second author. We show that the natural analogue of this result fails for general $α\in\overline{\mathbb Q}$ by giving an infinite family of quadratic counterexamples. As part of this construction, we provide an explicit formula to compute $M_t(D^{1/k})$ for a square-free $D \in \mathbb N$. |
| title | The $t$-metric Mahler measures of surds and rational numbers |
| topic | Number Theory 11R04, 11R09 (Primary), 11C08, 12E05 (Secondary) |
| url | https://arxiv.org/abs/1408.4166 |