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YCor
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Given a non increasing function $\psi$ the $\psi$ approximable points in $\mathbb{R}^n$ is defined as $W(\psi)=\{x\in\mathbb{R}^n:|qx-p|<\psi(q)\}$ for infinitely many $(q,p)\in \mathbb{Z}^m\times\mathbb{Z}$. Now then one defines exact $\psi$ approximable vector as $E(\psi)=W(\psi)\setminus\cup_{c<1} W(c\psi)$. I had read many results concerning $E(\psi)$ in classical setting. But I am unable to find any result concerning $E(\psi)$ say the Hausdorff dimension of the set in the setting of field of formal series or even $p$ adic field. I am just curious to know whether the exact approximable notion in these fields as norm is discrete somehow gives some trivial things.

There are results concerning size of Badly approximable vectors in these settings but I couldn't find anything related to exact approximable vectors in these setting. Any reference if there are any will be extremely beneficial. Or if that is not interesting in $Q_p$ any comment regarding that also is also very helpful.

Given a non increasing function $\psi$ the $\psi$ approximable points in $\mathbb{R}^n$ is defined as

Given a non increasing function $\psi$ the $\psi$ approximable points in $\mathbb{R}^n$ is defined as $W(\psi)=\{x\in\mathbb{R}^n:|qx-p|<\psi(q)\}$ for infinitely many $(q,p)\in \mathbb{Z}^m\times\mathbb{Z}$. Now then one defines exact $\psi$ approximable vector as $E(\psi)=W(\psi)\setminus\cup_{c<1} W(c\psi)$. I had read many results concerning $E(\psi)$ in classical setting. But I am unable to find any result concerning $E(\psi)$ say the Hausdorff dimension of the set in the setting of field of formal series or even $p$ adic field. I am just curious to know whether the exact approximable notion in these fields as norm is discrete somehow gives some trivial things.

There are results concerning size of Badly approximable vectors in these settings but I couldn't find anything related to exact approximable vectors in these setting. Any reference if there are any will be extremely beneficial. Or if that is not interesting in $Q_p$ any comment regarding that also is also very helpful.

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User5
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Given a non increasing function $\psi$ the $\psi$ approximable points in $\mathbb{R}^n$ is defined as $W(\psi)=\{x\in\mathbb{R}^n:|qx-p|<\psi(q)\}$ for infinitely many $(q,p)\in \mathbb{Z}^m\times\mathbb{Z}$. Now then one defines exact $\psi$ approximable vector as $E(\psi)=W(\psi)\setminus\cup_{c<1} W(c\psi)$. I had read many results concerning $E(\psi)$ in classical setting. But I am unable to find any result concerning $E(\psi)$ say the Hausdorff dimension of the set in the setting of field of formal series or even $p$ adic field. I am just curious to know whether the exact approximable notion in these fields as norm is discrete somehow gives some trivial things.

There are results concerning size of Badly approximable vectors in these settings but I couldn't find anything related to exact approximable vectors in these setting. Any reference if there are any will be extremely beneficial. Or if that is not interesting in $Q_p$ any comment regarding that also is also very helpful.

Given a non increasing function $\psi$ the $\psi$ approximable points in $\mathbb{R}^n$ is defined as $W(\psi)=\{x\in\mathbb{R}^n:|qx-p|<\psi(q)\}$ for infinitely many $(q,p)\in \mathbb{Z}^m\times\mathbb{Z}$. Now then one defines exact $\psi$ approximable vector as $E(\psi)=W(\psi)\setminus\cup_{c<1} W(c\psi)$. I had read many results concerning $E(\psi)$ in classical setting. But I am unable to find any result concerning $E(\psi)$ say the Hausdorff dimension of the set in the setting of field of formal series or even $p$ adic field. I am just curious to know whether the exact approximable notion in these fields as norm is discrete somehow gives some trivial things.

There are results concerning size of Badly approximable vectors in these settings but I couldn't find anything related to exact approximable vectors in these setting. Any reference if there are any will be extremely beneficial. Or if that is not interesting in $Q_p$ any comment regarding that also is also very helpful.

Given a non increasing function $\psi$ the $\psi$ approximable points in $\mathbb{R}^n$ is defined as

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User5
User5
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Exact approximation in $p$ adic

Given a non increasing function $\psi$ the $\psi$ approximable points in $\mathbb{R}^n$ is defined as $W(\psi)=\{x\in\mathbb{R}^n:|qx-p|<\psi(q)\}$ for infinitely many $(q,p)\in \mathbb{Z}^m\times\mathbb{Z}$. Now then one defines exact $\psi$ approximable vector as $E(\psi)=W(\psi)\setminus\cup_{c<1} W(c\psi)$. I had read many results concerning $E(\psi)$ in classical setting. But I am unable to find any result concerning $E(\psi)$ say the Hausdorff dimension of the set in the setting of field of formal series or even $p$ adic field. I am just curious to know whether the exact approximable notion in these fields as norm is discrete somehow gives some trivial things.

There are results concerning size of Badly approximable vectors in these settings but I couldn't find anything related to exact approximable vectors in these setting. Any reference if there are any will be extremely beneficial. Or if that is not interesting in $Q_p$ any comment regarding that also is also very helpful.