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pro vyhledávání: '"Schleischitz, Johannes"'
Autor:
Schleischitz, Johannes
We refine upper bounds for the classical exponents of uniform approximation for a linear form on the Veronese curve in dimension from $3$ to $9$. For dimension three, this in particular shows that a bound previously obtained by two different methods
Externí odkaz:
http://arxiv.org/abs/2405.10086
Autor:
Schleischitz, Johannes
In the early 1900's, Maillet proved that the image of any Liouville number under a rational function with rational coefficients is again a Liouville number. The analogous result for quadratic Liouville matrices in higher dimension turns out to fail.
Externí odkaz:
http://arxiv.org/abs/2403.14434
We study the Folklore set of Dirichlet improvable matrices in $\mathbb R^{m\times n}$ which are neither singular nor badly approximable. We prove the non-emptiness for all positive integer pairs $m,n$ apart from $\{m,n\}=\{ 1,1\}$ and $\{m,n\}=\{ 2,3
Externí odkaz:
http://arxiv.org/abs/2402.13451
Autor:
Schleischitz, Johannes
Recently Koivusalo, Levesley, Ward and Zhang introduced the set of simultaneously $\Phi$-badly approximable real vectors of $\mathbb{R}^m$ with respect to an approximation function $\Phi$, and determined its Hausdorff dimension for the special class
Externí odkaz:
http://arxiv.org/abs/2312.14559
Autor:
Schleischitz, Johannes
We aim to fill a gap in the proof of an inequality relating two exponents of uniform Diophantine approximation stated in a paper by Bugeaud. We succeed to verify the inequality in several instances, in particular for small dimension. Moreover, we pro
Externí odkaz:
http://arxiv.org/abs/2312.08797
The Generalised Baker-Schmidt Problem (1970) concerns the Hausdorff measure of the set of $\psi$-approximable points on a nondegenerate manifold. Beresnevich-Dickinson-Velani (in 2006, for the homogeneous setting) and Badziahin-Beresnevich-Velani (in
Externí odkaz:
http://arxiv.org/abs/2308.14471
The Generalised Baker-Schmidt Problem (1970) concerns the Hausdorff $f$-measure of the set of $\Psi$-approximable points on a nondegenerate manifold. We refine and extend our previous work [Int. Math. Res. Not. IMRN 2021, no. 12, 8845--8867] in which
Externí odkaz:
http://arxiv.org/abs/2302.10378
Autor:
Schleischitz, Johannes
Consider the integer best approximations of a linear form in $n\ge 2$ real variables. While it is well-known that any tail of this sequence always spans a lattice is sharp for any $n\ge 2$. In this paper, we determine the exact Hausdorff and packing
Externí odkaz:
http://arxiv.org/abs/2302.08403
Autor:
Schleischitz, Johannes
Publikováno v:
Q.J. Math. 75 (2024), no. 1, 11-29
For $m\ge 2$, consider $K$ the $m$-fold Cartesian product of the limit set of an IFS of two affine maps with rational coefficients. If the contraction rates of the IFS are reciprocals of integers, and $K$ does not degenerate to singleton, we construc
Externí odkaz:
http://arxiv.org/abs/2210.07742
Autor:
Schleischitz, Johannes
Publikováno v:
Bull. Lond. Math. Soc. 55 (2023), no. 3, 1330-1339
For $n\geq 2$, we determine the Dirichlet spectrum in $\Rn$ with respect to a linear form and the maximum norm as the entire interval $[0,1]$. This natural result improves on recent work of Beresnevich, Guan, Marnat, Ram\'irez and Velani, and complem
Externí odkaz:
http://arxiv.org/abs/2205.10050