arXiv subjects

Johannes Schleischitz

Publications and source records attributed to Johannes Schleischitz.

At least 37 records · Page 2Linked to original sources

arXiv2022

Metric results on inhomogeneously singular vectors

Given $\ut\in\Rm$ and any norm $\Vert.\Vert$ on $\Rm$, we consider "inhomogeneously singular" vectors in $\Rm$ that admit an integer vector solution $(q,\underline{p})=(q,p_1,\ldots,p_m)$ to the system \[ 1\leq q\leq Q, \qquad \Vert q\ux-\underline{p}-\underlineθ\Vert\leq cQ^{-1/m} \] for any $c>0$ and all large $Q$. We show that this set has large packing dimension, and in view of recent deep results by Das, Fishman, Simmons, Urbański, our lower bounds are almost sharp (up to $O(m^{-1})$). We establish slightly weaker bounds for the Hausdorff dimension as well. Our bounds are applicable to the $b$-ary setting, i.e. when restricting to $q$ above integral powers of some given base $b\geq 2$. We further derive similar results for vectors on certain $m$-dimensional fractals, thereby contributing to a question of Bugeaud, Cheung and Chevallier and complementing recent work by Kleinbock, Moshchevitin and Weiss and by Khalil. Moreover, we show that in contrast to Liouville vectors, the set of singular vectors in $\Rm$ does not form a comeagre set. We infer this from a general new result that any comeagre set in $\Rm$ has full packing dimension. As another independent consequence of our method, we show that there are subsets $A,B$ of $\mathbb{R}^m$ for which $A+B=B+B=\mathbb{R}^m$ but $A+B$ has Hausdorff dimension less than $m$, and generalizations. The proofs rely on observations on sumsets and a result by Tricot involving Cartesian products and are surprisingly elementary. The topological results further use an observation of Erdős.

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arXiv2021

Applications of Siegel's Lemma to a system of linear forms and its minimal points

Consider a real matrix $Θ$ consisting of rows $(θ_{i,1},\ldots,θ_{i,n})$, for $1\leq i\leq m$. The problem of making the system linear forms $x_{1}θ_{i,1}+\cdots+x_{n}θ_{i,n}-y_{i}$ for integers $x_{j},y_{i}$ small naturally induces an ordinary and a uniform exponent of approximation, denoted by $w(Θ)$ and $\widehat{w}(Θ)$ respectively. For $m=1$, a sharp lower bound for the ratio $w(Θ)/\widehat{w}(Θ)$ was recently established by Marnat and Moshchevitin. We give a short, new proof of this result upon a hypothesis on the best approximation integer vectors associated to $Θ$. Our conditional result extends to general $m>1$ (but may not be optimal in this case). Moreover, our hypothesis is always satisfied in particular for $m=1, n=2$ and thereby unconditionally confirms a previous observation of Jarník. We formulate our results in the more general context of approximation of subspaces of Euclidean spaces by lattices. We further establish criteria upon which a given number $\ell$ of consecutive best approximation vectors are linearly independent. Our method is based on Siegel's Lemma.

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arXiv2021

Classical and uniform exponents of multiplicative $p$-adic approximation

Let $p$ be a prime number and $ξ$ an irrational $p$-adic number. Its irrationality exponent $μ(ξ)$ is the supremum of the real numbers $μ$ for which the system of inequalities $$ 0 < \max\{|x|, |y|\} \le X, \quad |y ξ- x|_{p} \leq X^{-\hmu} $$ has a solution in integers $x, y$ for arbitrarily large real number $X$. Its multiplicative irrationality exponent $\tmu (ξ)$ (resp., uniform multiplicative irrationality exponent $\htmu (ξ)$) is the supremum of the real numbers $\hmu$ for which the system of inequalities $$ 0 < |x y|^{1/2} \le X, \quad |y ξ- x|_{p} \leq X^{-\hmu} $$ has a solution in integers $x, y$ for arbitrarily large (resp., for every sufficiently large) real number $X$. It is not difficult to show that $μ(ξ) \le \tmu(ξ) \le 2 μ(ξ)$ and $\htmu (ξ) \le 4$. We establish that the ratio between the multiplicative irrationality exponent $\tmu$ and the irrationality exponent $μ$ can take any given value in $[1, 2]$. Furthermore, we prove that $\htmu (ξ) \le (5 + \sqrt{5})/2$ for every $p$-adic number $ξ$.

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arXiv2021

Going-up theorems for simultaneous Diophantine approximation

We establish several new inequalities linking classical exponents of Diophantine approximation associated to a real vector $\underlineξ=(ξ,ξ^{2},\ldots,ξ^{N})$, in various dimensions $N$. We thereby obtain variants, and partly refinements, of recent results of Badziahin and Bugeaud. We further implicitly recover inequalities of Bugeaud and Laurent as special cases, with new proofs. Similar estimates concerning general real vectors (not on the Veronese curve) with $\mathbb{Q}$-linearly independent coordinates are addressed as well. Our method is based on Minkowski's Second Convex Body Theorem, applied in the framework of parametric geometry of numbers introduced by Schmidt and Summerer. We also frequently employ Mahler's Duality result on polar convex bodies.

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arXiv2021

Optimality of two inequalities for exponents of Diophantine approximation

We investigate two inequalities of Bugeaud and Laurent, each involving triples of classical exponents of Diophantine approximation associated to $\ux\in\mathbb{R}^n$. We provide a complete description of parameter triples that admit equality for suitable $\ux$, which turns out rather surprising. For $n=2$ our results agree with work of Laurent. Moreover, we establish lower bounds for the Hausdorff and packing dimensions of the involved $\ux$, and in special cases we can show they are sharp. Proofs are based on the variational principle in parametric geometry of numbers, we enclose sketches of associated combined graphs (templates) where equality is feasible. A twist of our construction provides refined information on the joint spectrum of the respective exponent triples.

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arXiv2021

On simultaneous rational approximation to a $p$-adic number and its integral powers, II

Let $p$ be a prime number. For a positive integer $n$ and a real number $ξ$, let $λ_n (ξ)$ denote the supremum of the real numbers $λ$ for which there are infinitely many integer tuples $(x_0, x_1, \ldots , x_n)$ such that $| x_0 ξ- x_1|_p, \ldots , | x_0 ξ^n - x_n|_p$ are all less than $X^{-λ- 1}$, where $X$ is the maximum of $|x_0|, |x_1|, \ldots , |x_n|$. We establish new results on the Hausdorff dimension of the set of real numbers $ξ$ for which $λ_n (ξ)$ is equal to (or greater than or equal to) a given value.

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arXiv2021

On geometry of numbers and uniform rational approximation to the Veronese curve

Consider the classical problem of rational simultaneous approximation to a point in $\mathbb{R}^{n}$. The optimal lower bound on the gap between the induced ordinary and uniform approximation exponents has been established by Marnat and Moshchevitin in 2018. Recently Nguyen, Poels and Roy provided information on the best approximating rational vectors to points where the gap is close to this minimal value. Combining the latter result with parametric geometry of numbers, we effectively bound the dual linear form exponents in the described situation. As an application, we slightly improve the upper bound for the classical exponent of uniform Diophantine approximation $\widehatλ_{n}(ξ)$, for even $n\geq 4$. Unfortunately our improvements are small, for $n=4$ only in the fifth decimal digit. However, the underlying method in principle can be improved with more effort to provide better bounds. We indeed establish reasonably stronger results for numbers which almost satisfy equality in the estimate by Marnat and Moshchevitin. We conclude with consequences on the classical problem of approximation to real numbers by algebraic numbers/integers of uniformly bounded degree.

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arXiv2019

An improved bound in Wirsing's problem

We improve the lower bound for the classical exponent of approximation $w_{n}^{\ast}(ξ)$ connected to Wirsing's famous problem of approximation to real numbers by algebraic numbers of degree at most $n$. Our bound exceeds $n/\sqrt{3}\approx 0.5773n$ and thus provides a reasonable qualitative improvement to previous bounds of order $n/2+O(1)$. We further establish new relations between several classical exponents of approximation.

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arXiv2019

On intrinsic and extrinsic rational approximation to Cantor sets

We establish various new results on a problem proposed by K. Mahler in 1984 concerning rational approximation to fractal sets by rational numbers inside and outside the set in question, respectively. Some of them provide a natural continuation and improvement of recent results of Broderick, Fishman and Reich and Fishman and Simmons. A key feature is that many of our new results apply to more general, multi-dimensional fractal sets and require only mild assumptions on the iterated function system. Moreover we provide a non-trivial lower bound for the distance of a rational number $p/q$ outside the Cantor middle third set $C$ to the set $C$, in terms of the denominator $q$. We further discuss patterns of rational numbers in fractal sets. We want to highlight two of them: Firstly, an upper bound for the number of rational (algebraic) numbers in a fractal set up to a given height (and degree) for a wide class of fractal sets. Secondly we find properties of the denominator structure of rational points in ''missing digit'' Cantor sets, generalizing claims of Nagy and Bloshchitsyn.

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arXiv2019

The generalised Baker--Schmidt problem on hypersurfaces

The Generalised Baker--Schmidt Problem (1970) concerns the $f$-dimensional Hausdorff measure of the set of $ψ$-approximable points on a nondegenerate manifold. There are two variants of this problem, concerning simultaneous and dual approximation. 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 corresponding convergence counterpart represents a major challenging open question and the progress thus far has only been attained over planar curves. In this paper, we settle this problem for hypersurfaces in a more general setting, i.e. for inhomogeneous approximations and with a non-monotonic multivariable approximating function.

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arXiv2019

On a question of Schmidt and Summerer concerning $3$-systems

Following a suggestion of W.M. Schmidt and L. Summerer, we construct a proper $3$-system $(P_{1},P_{2},P_{3})$ with the property $\overlineφ_{3}=1$. In fact, our method generalizes to provide $n$-systems with $\overlineφ_{n}=1$, for arbitrary $n\geq 3$. We visualize our constructions with graphics. We further present explicit examples of numbers $ξ_{1}, \ldots, ξ_{n-1}$ that induce the $n$-systems in question.

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arXiv2019

Uniform Diophantine approximation and best approximation polynomials

Let $ζ$ be a real transcendental number. We introduce a new method to find upper bounds for the classical exponent $\widehat{w}_{n}(ζ)$ concerning uniform polynomial approximation. Our method is based on the parametric geometry of numbers introduced by Schmidt and Summerer, and transference of the original approximation problem in dimension $n$ to suitable higher dimensions. For large $n$, we can provide an unconditional bound of order $\widehat{w}_{n}(ζ)\leq 2n-2+o(1)$. While this improves the bound of order $2n-\frac{3}{2}+o(1)$ due to Bugeaud and the author, it is unfortunately slightly weaker than what can be obtained when incorporating a recently proved conjecture of Schmidt and Summerer. However, the method also enables us to establish significantly stronger conditional bound upon a certain presumably weak assumption on the structure of the best approximation polynomials. Thereby we provide serious evidence that the exponent should be reasonably smaller than the known upper bounds.

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arXiv2018

Construction of numbers with almost all convergents in a Cantor set

In 1984, K. Mahler asked how well elements in the Cantor middle third set can be approximated by rational numbers from that set, and by rational numbers outside of that set. We consider more general missing digit sets $C$ and construct numbers in $C$ that are arbitrarily well approximable by rationals in $C$, but badly approximable by rationals outside of $C$. More precisely, we construct them so that all but finitely many of their convergents lie in $C$.

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arXiv2017

On the discrepancy between best and uniform approximation

For $ζ$ a transcendental real number, we consider the classical Diophantine exponents $w_{n}(ζ)$ and $\widehat{w}_{n}(ζ)$. They measure how small $| P(ζ)|$ can be for an integer polynomial $P$ of degree at most $n$ and naive height bounded by $X$, for arbitrarily large and all large $X$, respectively. The discrepancy between the exponents $w_{n}(ζ)$ and $\widehat{w}_{n}(ζ)$ has attracted interest recently. Studying parametric geometry of numbers, W. Schmidt and L. Summerer were the first to refine the trivial inequality $w_{n}(ζ)\geq \widehat{w}_{n}(ζ)$. Y. Bugeaud and the author found another estimation provided that the condition $w_{n}(ζ)>w_{n-1}(ζ)$ holds. In this paper we establish an unconditioned version of the latter result, which can be regarded as a proper extension. Unfortunately, the new contribution involves an additional exponent and is of interest only in certain cases.

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arXiv2017

Diophantine approximation in prescribed degree

We investigate approximation to a given real number by algebraic numbers and algebraic integers of prescribed degree. We deal with both best and uniform approximation, and highlight the similarities and differences compared with the intensely studied problem of approximation by algebraic numbers (and integers) of bounded degree. We establish the answer to a question of Bugeaud concerning approximation to transcendental real numbers by quadratic irrational numbers, and thereby we refine a result of Davenport and Schmidt from 1967. We also obtain several new characterizations of Liouville numbers, and certain new insights on inhomogeneous Diophantine approximation. As an auxiliary side result, we provide an upper bound for the number of certain linear combinations of two given relatively prime integer polynomials with a linear factor. We conclude with several open problems.

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arXiv2017

An equivalence principle between polynomial and simultaneous Diophantine approximation

We show that Mahler's classification of real numbers $ζ$ with respect to the growth of the sequence $(w_{n}(ζ))_{n\geq 1}$ is equivalently induced by certain natural assumptions on the decay of the sequence $(λ_{n}(ζ))_{n\geq 1}$ concerning simultaneous rational approximation. Thereby we obtain a much clearer picture on simultaneous approximation to successive powers of a real number in general. Another variant of the Mahler classification concerning uniform approximation by algebraic numbers is shown as well. Our method has several applications to classic exponents of Diophantine approximation and metric theory. We deduce estimates on the Hausdorff dimension of well-approximable vectors on the Veronese curve and refine the best known upper bound for the exponent $\widehatλ_{n}(ζ)$ for even $n\geq 4$.

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arXiv2017

Cubic approximation to Sturmian continued fractions

We determine the classical approximation constants $w_{3}(ζ),w_{3}^{\ast}(ζ),λ_{3}(ζ)$ such as the uniform constants $\widehat{w}_{3}(ζ),\widehat{w}_{3}^{\ast}(ζ),\widehatλ_{3}(ζ)$ associated to real numbers $ζ$ whose continued fraction expansions are given by a Sturmian word. We more generally provide a description of the combined graph of the parametric successive minima functions defined by Schmidt and Summerer in dimension three for such Sturmian continued fractions. This both complements similar results due to Bugeaud and Laurent concerning the two-dimensional constants and generalizes a recent result of the author. As a side result we obtain new information on the spectra of certain exponents of approximation. Moreover, we provide some information on the exponents $λ_{n}(ζ)$ for a Sturmian continued fraction $ζ$ and arbitrary $n$.

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arXiv2017

Some notes on the regular graph defined by Schmidt and Summerer and uniform approximation

Within the study of parametric geometry of numbers W. Schmidt and L. Summerer introduced so-called regular graphs. Roughly speaking the successive minima functions for the classical simultaneous Diophantine approximation problem have a very special pattern if the vector $\underlineζ$ induces a regular graph. The regular graph is in particular of interest due to a conjecture by Schmidt and Summer concerning classic approximation constants. This paper aims to provide several new results on the behavior of the successive minima functions for the regular graph. Moreover, we improve the best known upper bounds for the classic approximation constants $\hat{w}_{n}(ζ)$, valid uniformly for all transcendental $ζ$, provided that the Schmidt-Summerer conjecture is true.

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