Bad approximability, bounded ratios and Diophantine exponents
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912869543575552 |
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| author | Marnat, Antoine Moshchevitin, Nikolay Schleischitz, Johannes |
| author_facet | Marnat, Antoine Moshchevitin, Nikolay Schleischitz, Johannes |
| contents | For a real $m\times n$ matrix $\pmbξ$, we consider its sequence of best Diophantine approximation vectors $ \pmb{x}_i \in \mathbb{Z}^n, \, i =1,2,3, ... $, the sequences of its norms $X_i = \|\pmb{x}_i\|$ and the norms of remainders $L_i = \|\pmbξ\pmb{x}_i\|$. It is known that, in the cases $m=1$, bad approximability of $\pmbξ$ is equivalent to the boundedness of ratios $\frac{X_{i+1}}{X_i}$, while for $n=1$ bad approximability of $\pmbξ$ is equivalent to the boundedness of ratios $ \frac{L_i}{L_{i+1}}$. Moreover, carefully constructed example show that in the cases $m=1$ and $n=1$ boundedness of ratios $ \frac{L_i}{L_{i+1}}$ and $\frac{X_{i+1}}{X_i}$ respectively (the order of ratios changed), does not imply bad approximability of $\pmbξ$. In the present paper, we study the impact of the boundedness of ratios on Diophantine properties of $\pmbξ$, in particular, what restrictions it gives for Diophantine exponents $ω(\pmbξ)$ and $\hatω(\pmbξ)$. One of our particular results deals with the case $m=n=2$. We prove that for $2\times 2 $ matrices $\pmbξ$ boundedness of both ratios $ \frac{X_{i+1}}{X_i}, \frac{L_i}{L_{i+1}} $ implies inequality $\hatω(\pmbξ)\le \frac{4}{3}$ and that this result is optimal. Our methods combine parametric geometry of numbers as well as more classical tools. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_15964 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bad approximability, bounded ratios and Diophantine exponents Marnat, Antoine Moshchevitin, Nikolay Schleischitz, Johannes Number Theory 11J13 For a real $m\times n$ matrix $\pmbξ$, we consider its sequence of best Diophantine approximation vectors $ \pmb{x}_i \in \mathbb{Z}^n, \, i =1,2,3, ... $, the sequences of its norms $X_i = \|\pmb{x}_i\|$ and the norms of remainders $L_i = \|\pmbξ\pmb{x}_i\|$. It is known that, in the cases $m=1$, bad approximability of $\pmbξ$ is equivalent to the boundedness of ratios $\frac{X_{i+1}}{X_i}$, while for $n=1$ bad approximability of $\pmbξ$ is equivalent to the boundedness of ratios $ \frac{L_i}{L_{i+1}}$. Moreover, carefully constructed example show that in the cases $m=1$ and $n=1$ boundedness of ratios $ \frac{L_i}{L_{i+1}}$ and $\frac{X_{i+1}}{X_i}$ respectively (the order of ratios changed), does not imply bad approximability of $\pmbξ$. In the present paper, we study the impact of the boundedness of ratios on Diophantine properties of $\pmbξ$, in particular, what restrictions it gives for Diophantine exponents $ω(\pmbξ)$ and $\hatω(\pmbξ)$. One of our particular results deals with the case $m=n=2$. We prove that for $2\times 2 $ matrices $\pmbξ$ boundedness of both ratios $ \frac{X_{i+1}}{X_i}, \frac{L_i}{L_{i+1}} $ implies inequality $\hatω(\pmbξ)\le \frac{4}{3}$ and that this result is optimal. Our methods combine parametric geometry of numbers as well as more classical tools. |
| title | Bad approximability, bounded ratios and Diophantine exponents |
| topic | Number Theory 11J13 |
| url | https://arxiv.org/abs/2505.15964 |