arXiv subjects

Johannes Schleischitz

Publications and source records attributed to Johannes Schleischitz.

51 records · Page 3Linked to original sources

arXiv2017

Rational approximation to algebraic varieties and a new exponent of simultaneous approximation

This paper deals with two main topics related to Diophantine approximation. Firstly, we show that if a point on an algebraic variety is approximable by rational vectors to a sufficiently large degree, the approximating vectors must lie in the topological closure of the rational points on the variety. In many interesting cases, in particular if the set of rational points on the variety is finite, this closure does not exceed the set of rational points on the variety itself. This result enables easier proofs of several known results as special cases. The proof can be generalized in some way and encourages to define a new exponent of simultaneous approximation. The second part of the paper is devoted to the study of this exponent.

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arXiv2017

On uniform approximation to successive powers of a real number

We establish new inequalities involving classical exponents of Diophantine approximation. This allows for improving on the work of Davenport, Schmidt and Laurent concerning the maximum value of the exponent $\hatλ_{n}(ζ)$ among all real transcendental $ζ$. In particular we refine the estimation $\hatλ_{n}(ζ)\leq \lceil n/2\rceil^{-1}$ due to M. Laurent by $\hatλ_{n}(ζ)\leq \hat{w}_{\lceil n/2\rceil}(ζ)^{-1}$ for all $n\geq 1$, and for even $n$ we replace the bound $2/n$ for $\hatλ_{n}(ζ)$ first found by Davenport and Schmidt by roughly $\frac{2}{n}-\frac{1}{2n^{3}}$, which provides the currently best known bounds when $n\geq 6$.

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arXiv2016

Generalizations of a result of Jarnik on simultaneous approximation

Consider a non-increasing function $Ψ$ from the positive reals to the positive reals with decay $o(1/x)$ as $x$ tends to infinity. Jarnik proved in 1930 that there exist real numbers $ζ_{1},...,ζ_{k}$ together with $1$ linearly independent over $\mathbb{Q}$ with the property that all $qζ_{j}$ have distance to the nearest integer smaller than $Ψ(q)$ for infinitely many positive integers $q$, but not much smaller in a very strict sense. We give an effective generalization of this result to the case of successive powers of real $ζ$. The method also allows to generalize corresponding results for $ζ$ contained in special fractal sets such as the Cantor set.

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arXiv2016

Approximation to an extremal number, its square and its cube

We study rational approximation properties for successive powers of extremal numbers defined by Roy. For $n\in{\{1,2\}}$, the classic approximation constants $λ_{n}(ζ),\hatλ_{n}(ζ),w_{n}(ζ),\hat{w}_{n}(ζ)$ connected to an extremal number $ζ$ have been established and in fact much more is known. However, so far almost nothing had been known for $n\geq 3$. In this paper we determine all classic approximation constants as above for $n=3$. Our methods will more generally provide detailed information on the combined graph defined by Schmidt and Summerer assigned to an extremal number, its square and its cube. We provide some results for $n=4$ as well. In the course of the proof of the main results we establish a very general connection between Khintchine's transference inequalities and uniform approximation.

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arXiv2016

Integral powers of numbers in small intervals modulo $1$: The cardinality gap phenomenon

This paper deals with the distribution of $αζ^{n} \bmod 1$, where $α\neq 0,ζ>1$ are fixed real numbers and $n$ runs through the positive integers. Denote by $\Vert.\Vert$ the distance to the nearest integer. We investigate the case of $αζ^{n}$ all lying in prescribed small intervals modulo $1$ for all large $n$, with focus on the case $\Vertαζ^{n}\Vert \leq ε$ for small $ε>0$. We are particularly interested in what we call cardinality gap phenomena. For example for fixed $ζ>1$ and small $ε>0$ there are at most countably many values of $α$ such that $\Vertαζ^{n}\Vert \leq ε$ for all large $n$, whereas larger $ε$ induces an uncountable set. We investigate the value of $ε$ at which the gap occurs. We will pay particular attention to the case of algebraic and, more specific, rational $ζ>1$. Results concerning Pisot and Salem numbers such as some contribution to Mahler's $3/2$-problem are implicitly deduced. We study similar questions for fixed $α\neq 0$ as well.

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arXiv2016

On uniform approximation to real numbers

Let $n \ge 2$ be an integer and $ξ$ a transcendental real number. We establish several new relations between the values at $ξ$ of the exponents of Diophantine approximation $w_n, w_{n}^{\ast}, \hat{w}_{n}$, and $\hat{w}_{n}^{\ast}$. Combining our results with recent estimates by Schmidt and Summerer allows us to refine the inequality $\hat{w}_{n}(ξ) \le 2n-1$ proved by Davenport and Schmidt in 1969.

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arXiv2016

On the spectrum of Diophantine approximation constants

The approximation constant $λ_{k}(ζ)$ is defined as the supremum of real $η$ such that $\Vert ζ^{j}x\Vert\leq x^{-η}$ for $1\leq j\leq k$ has infinitely many integer solutions $x$. Here $\Vert.\Vert$ denotes the distance to the closest integer. We establish a connection on the joint spectrum $(λ_{1}(ζ),λ_{2}(ζ),\ldots)$ which will lead to various improvements of known results on the individual spectrum of the approximation constants $λ_{k}(ζ)$ as well. In particular, this extends a result by Bugeaud to the case of arbitrary dimension $k$. Concretely, given $k\geq 1$ and $λ\geq 1$, we infer {\em explicit} constructions of $ζ$ in the Cantor set with $λ_{k}(ζ)=λ$.

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arXiv2016

On a problem posed by Mahler

E. Maillet proved that the set of Liouville numbers is preserved under rational functions with rational coefficients. Based on this result, a problem posed by Kurt Mahler is to investigate whether there exist entire transcendental functions with this property or not. For large parametrized classes of Liouville numbers, we construct such functions and moreover we show that it can be constructed such that all their derivatives share this property. We use a completely different approach than in a recent paper, where functions with a different invariant subclass of Liouville numbers were constructed (though with no information on derivatives). More generally, we study the image of Liouville numbers under analytic functions, with particular attention to $f(z)=z^{q}$ where $q$ is a rational number.

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arXiv2016

Rational approximation to surfaces defined by polynomials in one variable

We study the rational approximation properties of special manifolds defined by a set of polynomials with rational coefficients. Mostly we will assume the case of all polynomials to depend on only one variable. In this case the manifold can be viewed as a Cartesian product of polynomial curves and it is possible to generalize recent results concerning such curves with similar concepts. There is hope that the method leads to insights on how to treat more general manifolds defined by arbitrary polynomials with rational coefficients.

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arXiv2015

Diophantine approximation on polynomial curves

In a paper from 2010, Budarina, Dickinson and Levesley studied the rational approximation properties of curves parametrized by polynomials with integral coefficients in Euclidean space of arbitrary dimension. Assuming the dimension is at least three and excluding the case of linear dependence of the polynomials together with $P(X)\equiv 1$ over the rational number field, we establish proper generalizations of their main result.

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arXiv2015

On a $\mathbb{Z}$-module connected to approximation theory

This paper deals with the set of $α\in{\mathbb{R}}$ such that $αζ^{n} \bmod 1$ tends to $0$ for a fixed $ζ\in{\mathbb{R}}$, which we call $\mathscr{M}_ζ$. Predominately the case of Pisot numbers $ζ$ is studied. In this case the inclusions $\mathcal{O}_{\mathbb{Q}(ζ)}\subset\mathscr{M}_ζ\subset\mathbb{Q}(ζ)$ are known. We will show the properties of $\mathscr{M}_ζ$ are connected to the module structure of the ring of integers $\mathcal{O}_{\mathbb{Q}(ζ)}$. We will describe the module structure of $\mathscr{M}_ζ$ and how much $\mathscr{M}_ζ$ differs from $\mathcal{O}_{\mathbb{Q}(ζ)}$. The results besides allow to give some information on the shape of integral bases of real number fields.

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arXiv2015

On Approximation constants for Liouville numbers

We investigate some Diophantine approximation constants related to the simultaneous approximation of $(ζ,ζ^{2},\ldots,ζ^{k})$ for Liouville numbers $ζ$. For a certain class of Liouville numbers including the famous representative $\sum_{n\geq 1} 10^{-n!}$ and numbers in the Cantor set, we explicitly determine all approximation constants simultaneously for all $k\geq 1$.

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arXiv2014

On the rate of accumulation of $αζ^{n}$ mod 1 to 0

In this paper we study the distribution of the sequence $(αζ^{n})_{n\geq 1}$ mod $1$, where $α,ζ$ are fixed positive real numbers, with special focus on the accumulation point $0$. For this purpose we introduce approximation constants $\underlineσ(α,ζ),\overlineσ(α, ζ)$ and study their properties in dependence of $α,ζ$, distinguishing in particular the cases of Pisot numbers, algebraic non Pisot numbers and transcendental values of $α$ as well as $ζ$.

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arXiv2013

Two estimates concerning classical Diophantine approximation constants

In this paper we aim to prove two inequalities involving the classical approximation constants $w_{n}^{\prime}(ζ),\hat{w}_{n}^{\prime}(ζ)$ that stem from the simultaneous approximation problem $|ζ^{j}x-y_{j}|$, $1\leq j\leq n$, on the one side and the constants $w_{n}^{\ast}(ζ),\hat{w}_{n}^{\ast}(ζ)$ connected to approximation with algebraic numbers of degree $\leq n$ on the other side. We concretely prove $w_{n}^{\ast}(ζ)\hat{w}_{n}^{\prime}(ζ)\geq 1$ and $\hat{w}_{n}^{\ast}(ζ)w_{n}^{\prime}(ζ)\geq 1$. The first result is due to W. Schmidt, however our method of proving it allows to derive the other inequality as a dual result. Finally we will discuss estimates of $w_{n}^{\ast}(ζ), \hat{w}_{n}^{\ast}(ζ)$ uniformly in $ζ$ depending only on $n$ as an application.

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arXiv2013

Diophantine approximation and special Liouville numbers

This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $ζ_{1},ζ_{2},...,ζ_{k}$. The approach relies on results on the connection between the set of all $s$-adic expansions ($s\geq 2$) of $ζ_{1},ζ_{2},...,ζ_{k}$ and their associated approximation constants. As an application, explicit construction of real numbers $ζ_{1},ζ_{2},...,ζ_{k}$ with prescribed approximation properties are deduced and illustrated by Matlab plots.

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