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- W4226245671 abstract "In this paper, we study inhomogeneous Diophantine approximation over the completion $K_v$ of a global function field $K$ (over a finite field) for a discrete valuation $v$, with affine algebra $R_v$. We obtain an effective upper bound for the Hausdorff dimension of the set [ mathbf{Bad}_A(epsilon)=left{boldsymbol{theta}in K_v^{,m} : liminf_{(mathbf{p},mathbf{q})in R_v^{,m} times R_v^{,n}, |mathbf{q}|to infty} |mathbf{q}|^n |Amathbf{q}-boldsymbol{theta}-mathbf{p}|^m geq epsilon right}, ] of $epsilon$-badly approximable targets $boldsymbol{theta}in K_v^{,m}$ for a fixed matrix $Ainmathscr{M}_{m,n}(K_v)$, using an effective version of entropy rigidity in homogeneous dynamics for an appropriate diagonal action on the space of $R_v$-grids. We further characterize matrices $A$ for which $mathbf{Bad}_A(epsilon)$ has full Hausdorff dimension for some $epsilon>0$ by a Diophantine condition of singularity on average. Our methods also work for the approximation using weighted ultrametric distances." @default.
- W4226245671 created "2022-05-05" @default.
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- W4226245671 date "2021-12-08" @default.
- W4226245671 modified "2023-09-26" @default.
- W4226245671 title "On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields" @default.
- W4226245671 doi "https://doi.org/10.48550/arxiv.2112.04144" @default.
- W4226245671 hasPublicationYear "2021" @default.
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