arXiv subjects

Johannes Schleischitz

Publications and source records attributed to Johannes Schleischitz.

At least 19 recordsLinked to original sources

arXiv2026

The Cartesian product of exact approximation sets

We determine the Hausdorff and packing dimensions of Cartesian products of one-dimensional exact approximation sets. Our main result establishes the exact-approximation counterpart of the recent product theorem of Wang and Wu (2024) for limsup approximation sets, showing that passing to the substantially smaller exact approximation sets (liminf sets) does not reduce the Hausdorff dimension of the Cartesian product. One of the key ingredients is a refinement of the well-distributed-system framework of Bandi--Ghosh--Nandi (2023) by exploiting the fine arithmetic distribution of rational points which then gives the Hausdorff dimension of the product set under a weaker convergence condition.

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arXiv2026

Bad approximability, bounded ratios and Diophantine exponents

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.

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arXiv2026

On an inhomogeneous uniform Littlewood type problem

We show that a fully inhomogeneous uniform Littlewood type problem has a negative answer and the counterexamples form a dense $G_δ$ set. This extends the author's recent analogous result for the homogeneous case. The main difficulty in the general setting is the semi-homogeneous case where one linear form is homogeneous and the other inhomogeneous with irrational shift. We further address the higher dimensional analogue.

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arXiv2026

Disproof of the uniform Littlewood conjecture

We show that the uniform Littlewood Conjecture (ULC) recently introduced by Bandi, Fregoli and Kleinbock is false. More precisely the counterexamples form a residual set, the method further suggests positive Hausdorff dimension. For a mildly twisted problem, we indeed separately show that the Hausdorff dimension is at least $1$. Moreover, we disprove a uniform version of the $p$-adic Littlewood problem, as well as some twisted weaker version of a more general $S$-arithmetic setting, for any proper subset (possible infinite) of primes $S$. The latter contrasts the classical (non-uniform) case where the answer is known to be affirmative when $S$ has at least two elements. The disproof of ULC, our main new result, is semi-constructive; the non-constructive part involves effective results on Zaremba's famous conjecture by Bourgain and Kontorovich, as well as estimates for the cardinality of product sets over finite fields.

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arXiv2025

On the Folklore set and Dirichlet spectrum for matrices

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 $\{m,n\}=\{ 2,3\}$ in a constructive manner. For a wide range of integer pairs $(m,n)$ we construct subsets of the Folklore set with an exact prescribed Dirichlet constant (in some right neighbourhood of $0$). This enables us to provide information on the Dirichlet Spectrum of matrices. The key technique of our construction is to build first vectors of a given Diophantine type, and then to show that most `liftings' to matrices will preserve this Diophantine type. This is a variant of a method introduced by Moshchevitin for uniform approximation. Our technique is often also applicable to arbitrary norms. As a corollary, we obtain lower bounds on the Hausdorff dimension of these sets. These statements complement previous results of the middle-named author (Selecta Math. 2023), Beresnevich et. al. (Adv. Math. 2023), and Das et. al. (Adv. Math. 2024).

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arXiv2024

On two uniform exponents of approximation related to Wirsing's problem

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 provide counterexamples to generalizations, which contrasts the case of ordinary approximation where such phenomena do not occur. Our results contribute to the understanding of the discrepancy between small absolute values of a polynomial at a given real number and approximation to the number by algebraic numbers of absolutely bounded degree, a fundamental issue in the famous problem of Wirsing and variants.

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arXiv2024

Uniform dual approximation to Veronese curves in small dimension

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 is not sharp. Our proof involves parametric geometry of numbers and investigation of geometric properties of best approximation polynomials. Slightly stronger bounds have been obtained by Poels with a different method contemporarily. In fact, we obtain the same bounds as a conditional result.

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arXiv2024

On Wirsing's problem in small exact degree

We investigate a variant of Wirsing's problem on approximation to a real number by real algebraic numbers of degree exactly $n$. This has been studied by Bugeaud and Teulie. We improve their bounds for degrees up to $n=7$. Moreover, we obtain results regarding small values of polynomials and approximation to a real number by algebraic integers in small prescribed degree. The main ingredient are irreducibility criteria for integral linear combinations of coprime integer polynomials. Moreover, for cubic polynomials these criteria improve results of Győry on a problem of Szegedy.

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arXiv2024

Maillet's property and Mahler's Conjecture on Liouville numbers fail for matrices

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. In fact, using a result by Kleinbock and Margulis, we show that among analytic matrix functions in dimension $n\ge 2$, Maillet's invariance property is only true for Möbius transformations with special coefficients. This implies that the analogue in higher dimension of an open question of Mahler on the existence of transcendental entire functions with Maillet's property has a negative answer. On the other hand, extending a topological argument of Erdős, we prove that for any injective continuous self mapping on the space of rectangular matrices, many Liouville matrices are mapped to Liouville matrices. Dropping injectivity, we consider setups similar to Alniaçik and Saias and show that the situation depends on the matrix dimensions $m,n$. Finally we discuss extensions of a related result by Burger to quadratic matrices. We state several open problems along the way.

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arXiv2024

The set of $Φ$ badly approximable matrices has full Hausdorff dimension

Recently Koivusalo, Levesley, Ward and Zhang introduced the set of simultaneously $Φ$-badly approximable real vectors of $\mathbb{R}^m$ with respect to an approximation function $Φ$, and determined its Hausdorff dimension for the special class of power functions $Φ(t)=t^{-τ}$. We refine this by naturally extending the formula to arbitrary decreasing functions, in terms of the lower order of $1/Φ$ at infinity. We also provide an alternative, rather mild condition on $Φ$ for this conclusion. Moreover, our results apply in the general matrix setting, and we establish an according formula for packing dimension as well. Thereby we also complement a recent refinement by Bandi and de Saxcé on the smaller set of exact approximation with respect to $Φ$. Our basic tool is (a uniform variant of) the variational principle by Das, Fishman, Simmons, Urbański. We also prove some new lower estimates regarding the set of exact approximation order in the matrix setting, which are sharp in special instances. For this we combine the result by Bandi and de Saxcé with a method developed by Moshchevitin.

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arXiv2023

Dirichlet is not just bad and singular in many rational IFS fractals

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 construct vectors in $K$ that lie within the ``folklore set'' as defined by Beresnevich et al., meaning they are Dirichlet improvable but not singular or badly approximable (in fact our examples are Liouville vectors). We further address the topic of lower bounds for the Hausdorff and packing dimension of these folklore sets within $K$, however we do not compute bounds explicitly. Our class of fractals extends (Cartesian products of) classical missing digit fractals, for which analogous results had recently been obtained.

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arXiv2023

Dual $p$-adic Diophantine approximation on manifolds

The Generalised Baker-Schmidt Problem (1970) concerns the Hausdorff measure of the set of $ψ$-approximable points on a nondegenerate manifold. Beresnevich-Dickinson-Velani (in 2006, for the homogeneous setting) and Badziahin-Beresnevich-Velani (in 2013, for the inhomogeneous setting) proved the divergence part of this problem for dual approximation on arbitrary nondegenerate manifolds. The divergence part has also been resolved for the $p$-adic setting by Datta-Ghosh in 2022 for the inhomogeneous setting. The corresponding convergence counterpart represents a challenging open problem. In this paper, we prove the homogeneous $p$-adic convergence result for hypersurfaces of dimension at least three with some mild regularity condition, as well as for some other classes of manifolds satisfying certain conditions. We provide similar, slightly weaker results for the inhomogeneous setting. We do not restrict to monotonic approximation functions.

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arXiv2023

Metric results on sumsets and Cartesian products of classes of Diophantine sets

Erdős proved that any real number can be written as a sum, and a product, of two Liouville numbers. Motivated by these results, we study sumsets of classes of real numbers with prescribed (or bounded) irrationality exponents. We show that such sumsets turn out to be large in general, indeed almost every real number with respect to Lebesgue measure can be written as the sum of two numbers with sufficiently large prescribed irrationality exponents. In fact the Hausdorff dimension of the complement is small, and the result remains true if we impose considerably refined conditions on the orders of rational approximation (``exact approximation'' with respect to an approximation function). As an application, we show that in many cases the Hausdorff dimension of Cartesian products of sets with prescribed irrationality exponent exceeds the expected dimension, that is the sum of the single Hausdorff dimensions. We also address their packing dimensions. Similar results hold when restricting to classical missing digit Cantor sets, relative to its natural Cantor measure. In particular, we prove that the subset of numbers with prescribed large irrationality exponent has full packing dimension, i.e. the same packing dimension as the entire Cantor set. This complements some results on the Hausdorff dimension of these sets, which is an extensively studied topic in Diophantine approximation. Our proofs are based on ideas of Erdős, but vastly extend them.

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arXiv2023

Dirichlet spectrum for one linear form

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, Ramírez and Velani, and complements a subsequent paper by the author where the analogous result was proved for simultaneous approximation. Various generalisations that can be obtained by similar methods as in the latter paper are indicated. We believe that our results are an important step towards resolving the very open analogous problem for a general system of linear forms.

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arXiv2023

The Baker-Schmidt problem for dual approximation and some classes of functions

The Generalised Baker-Schmidt Problem (1970) concerns the Hausdorff $f$-measure of the set of $Ψ$-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 we settled the problem (for dual approximation) for hypersurfaces. We verify the GBSP for certain classes of nondegenerate submanifolds of codimension greater than $1$. Concretely, for codimension two or three, we provide examples of manifolds where the dependent variables can be chosen as quadratic forms. Our method requires the manifold to have even dimension at least the minimum of four and half the dimension of the ambient space. We conjecture that these restrictions on the dimension of the manifold are sufficient to provide similar examples in general.

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arXiv2023

Metrical results on the geometry of best approximations for a linear form

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 dimension of the set where equality occurs, in terms of $n$. Moreover, independently we show that there exist real vectors whose best approximations lie in a union of two two-dimensional sublattices of $\Z^{n+1}$. Our lattices jointly span a lattice of dimension three only, thereby leading to an alternative constructive proof of Moshchevitin's result. We determine the packing dimension and up to a small error term $O(n^{-1})$ also the Hausdorff dimension of the according set. Our method combines a new construction for a linear form in two variables $n=2$ with a result by Moshchevitin to amplify them. We further employ the recent variatonal principle and some of its consequences, as well as estimates for Hausdorff and packing dimensions of Cartesian products and fibers. Our method permits much freedom for the induced classical exponents of approximation.

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arXiv2022

Diophantine approximation on curves

Let $g$ be a dimension function. The Generalised Baker-Schmidt Problem (1970) concerns the $g$-dimensional Hausdorff measure ($\HH^g$-measure) of the set of $Ψ$-approximable points on nondegenerate manifolds. The problem relates the `size' of the set of $Ψ$-approximable points with the convergence or divergence of a certain series. In the dual approximation setting, the divergence case has been established by Beresnevich-Dickinson-Velani (2006) for any nondegenerate manifold. The convergence case, however, represents a major challenging open problem and progress thus far has been effectuated in limited cases only. In this paper, we discuss and prove several results on the $\HH^g$-measure on Veronese curves in any dimension $n$. As a consequence of one of our results, we generalize recent results of Pezzoni [Acta Arith. 193 (2020), no. 3, 269-281] regarding $n=3$. This improvement evolves from a deeper investigation on general irreducibility considerations applicable in arbitrary dimensions. We further investigate the $\HH^g$-measure for convergence on planar curves. We show that the monotonicity assumption on a multivariable approximating function cannot be removed for planar curves.

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arXiv2022

Exact uniform approximation and Dirichlet spectrum in dimension at least two

For $m\geq 2$, we determine the Dirichlet spectrum in $\Rm$ with respect to simultaneous approximation and the maximum norm as the entire interval $[0,1]$. This complements previous work of several authors, especially Akhunzhanov and Moshchevitin, who considered $m=2$ and Euclidean norm. We construct explicit examples of real Liouville vectors realizing any value in the unit interval. In particular, for positive values, they are neither badly approximable nor singular. Thereby we obtain a constructive proof of the main claim in a recent paper by Beresnevich, Guan, Marnat, Ramírez and Velani, who obtained a countable partition of $[0,1]$ into intervals with each having non-empty intersection with the Dirichlet spectrum. Our construction is flexible enough to show that the according set of vectors with prescribed Dirichlet exponent has large packing dimension and rather large Hausdorff dimension as well. We establish a more general result on exact uniform approximation, applicable to a wide class of approximating functions. Our constructive proofs are considerably shorter and less involved than previous work on the topic. By minor twists of our proof, we infer similar, slightly weaker results when restricting to a certain class of classical fractal sets or other norms. In an Appendix we address the situation of a linear form.

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