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Volker Ziegler

Publications and source records attributed to Volker Ziegler.

At least 19 recordsLinked to original sources
arXiv2026

Complete Resolution Of A Family Of Twisted Thue Equations

One of the first infinite families of Thue equations, $$F_n(X)=X^3 - (n-1) X^2Y - (n+2)XY^2 - Y^3 = \pm 1$$ for $n\in \mathbb{Z}$, was solved by Thomas in 1990. This family is associated to the simplest cubic fields $\mathbb{Q}(\lambda)$ of Shanks, where $\lambda$ is a root of $F_n(X,1)$. Levesque and Waldschmidt twisted the Thue equations by an exponent $t$ and looked at the equation $$N_{\mathbb{Q}(\lambda)/\mathbb{Q}}(X-\lambda^t Y)=\pm 1,$$ where $t\in \mathbb{Z}$ with $t\neq 0$. In this paper, we find all solutions $(X,Y,n,t)\in \mathbb{Z}^4$ with $n,t\in\mathbb{Z}$ and $t\neq 0$ to this family of twisted Thue equations, thereby answering a question of Levesque and Waldschmidt.

math.NT↗
arXiv2026

On integers that are representable as the sum of two units

Let $K$ be a number field of degree $D$ with maximal order $\mathcal{O}_K$. We show that under certain conditions on $K$, which in particular are always satisfied if $D$ is odd or if $D \geq 3$ and $K$ is primitive, the set of positive integers $N_K$ that can be expressed as a sum of two units in $\mathcal{O}_K^*$ is a finite effectively computable set. This result partially resolves an open problem posed by Tinkov\'{a}, Yatsyna, and the first author. We illustrate our method by explicitly computing $N_K$ for the smallest totally real quintic field $K$ with Galois group $S_5$.

math.NT↗
arXiv2026

Product of powers of distinct primes as sums of Fibonacci numbers

Let $F_n$ be the $n$-th Fibonacci number. In this paper, we study the Diophantine equation $F_n+F_m=p^xq^y$ in nonnegative integers $n\ge m$, $x$ and $y$, where $p$ and $q$ are fixed distinct prime numbers. We determine all pairs of primes $(q,p)$ with $q\le \min\{1000,p\}$ such that the above equation has at least two solutions $(x,y)$ (and corresponding $m,n$) in positive integers.

math.NT↗
arXiv2026

Rational integers as sums of units -- the quadratic case

How many natural numbers below $X$ can be written as a sum of $k$ units of the ring of integers of a given number field? We give the asymptotics as $X$ gets large for quadratic number fields. This solves a problem of Jarden and Narkiewicz from 2007 for quadratic number fields.

math.NT↗
arXiv2024

Key Focus Areas and Enabling Technologies for 6G

We provide a taxonomy of a dozen enabling network architectures, protocols, and technologies that will define the evolution from 5G to 6G. These technologies span the network protocol stack, different target deployment environments, and various perceived levels of technical maturity. We outline four areas of societal focus that will be impacted by these technologies, and overview several research directions that hold the potential to address the problems in these important focus areas.

cs.NI↗
arXiv2024

Automatic Mitigation of Dynamic Atmospheric Turbulence Using Optical Phase Conjugation for Coherent Free-Space Optical Communications

Coherent detection can provide enhanced receiver sensitivity and spectral efficiency in free-space optical (FSO) communications. However, turbulence can cause modal power coupling effects on a Gaussian data beam and significantly degrade the mixing efficiency between the data beam and a Gaussian local oscillator (LO) in the coherent detector. Optical phase conjugation (OPC) in a photorefractive crystal can "automatically" mitigate turbulence by: (a) recording a back-propagated turbulence-distorted probe beam, and (b) creating a phase-conjugate beam that has the inverse phase distortion of the medium as the transmitted data beam. However, previously reported crystal-based OPC approaches for FSO links have demonstrated either: (i) a relatively fast response time of 35 ms but at a relatively low data rate (e.g., <1 Mbit/s), or (ii) a relatively high data rate of 2-Gbit/s but at a slow response time (e.g., >60 s). Here, we report an OPC approach for the automatic mitigation of dynamic turbulence that enables both a high data rate (8 Gbit/s) data beam and a rapid (<5 ms) response time. For a similar data rate, this represents a 10,000-fold faster response time than previous reports, thereby enabling mitigation for dynamic effects. In our approach, the transmitted pre-distorted phase-conjugate data beam is generated by four-wave mixing in a GaAs crystal of three input beams: a turbulence-distorted probe beam, a Gaussian reference beam regenerated from the probe beam, and a Gaussian data beam carrying a high-speed data channel. We experimentally demonstrate our approach in an 8-Gbit/s quadrature-phase-shift-keying coherent FSO link through emulated dynamic turbulence. Our results show ~10-dB improvement in the mixing efficiency of the LO with the data beam under dynamic turbulence with a bandwidth of up to ~260 Hz (Greenwood frequency).

physics.optics↗
arXiv2024

On a conjecture of Levesque and Waldschmidt II

Related to Shank's notion of simplest cubic fields, the family of parametrised Diophantine equations, \[ x^3 - (n-1) x^2 y - (n+2) xy^2 - 1 = \left( x - λ_0 y\right) \left(x-λ_1 y\right) \left(x - λ_2 y\right) = \pm 1, \] was studied and solved effectively by Thomas and later solved completely by Mignotte. An open conjecture of Levesque and Waldschmidt states that taking these parametrised Diophantine equations and twisting them not only once but twice, in the sense that we look at \[ f_{n,s,t}(x,y) = \left( x - λ_0^s λ_1^t y \right) \left( x - λ_1^sλ_2^t y \right) \left( x - λ_2^sλ_0^t y \right) = \pm 1, \] retains a result similar to what Thomas obtained in the original or Levesque and Waldschidt in the once-twisted ($t = 0$) case; namely, that non-trivial solutions can only appear in equations where the parameters are small. We confirm this conjecture, given that the absolute values of the exponents $s, t$ are not too large compared to the base parameter $n$.

math.NT↗
arXiv2023

On Pillai's Problem involving Lucas sequences of the second kind

In this paper we consider the Diophantine equation $ V_n - b^m = c $ for given integers $ b,c $ with $ b \geq 2 $, whereas $ V_n $ varies among Lucas-Lehmer sequences of the second kind. We prove under some technical conditions that if the considered equation has at least three solutions $ (n,m) $, then there is an upper bound on the size of the solutions as well as on the size of the coefficients in the characteristic polynomial of $ V_n $.

math.NT↗
arXiv2023

On a conjecture of Levesque and Waldschmidt

One of the first parametrised Thue equations, $$\left| X^3 - (n-1)X^2 Y - (n+2) XY^2 - Y^3 \right| = 1,$$ over the integers was solved by E. Thomas in 1990. If we interpret this as a norm-form equation, we can write this as $$\left| N_{K/\mathbb{Q}}\left( X - λ_0 Y \right) \right| = \left| \left( X-λ_0 Y \right) \left( X-λ_1 Y \right) \left( X-λ_2 Y \right) \right| =1$$ if $λ_0, λ_1, λ_2$ are the roots of the defining irreducible polynomial, and $K$ the corresponding number field.\par\medskip Levesque and Waldschmidt twisted this norm-form equation by an exponential parameter $s$ and looked, among other things, at the equation $$\left| N_{K/\mathbb{Q}}\left( X - λ_0^s Y \right) \right| = 1.$$ They solved this effectively and conjectured that introducing a second exponential parameter $t$ and looking at $$\left| N_{K/\mathbb{Q}}\left( X - λ_0^sλ_1^t Y \right) \right| = 1$$ does not change the effective solvability. \par\medskip We want to partially confirm this, given that $$\min\left( \left| 2s-t \right|, \left| 2t-s \right|, \left| s+t \right| \right) > \varepsilon \cdot \max\left( \left|s\right|, \left|t\right| \right) > 2,$$ i.e. the two exponents do not almost cancel in specific cases.

math.NT↗
arXiv2023

On sums of two Fibonacci numbers that are powers of numbers with limited Hamming weight

In 2018, Luca and Patel conjectured that the largest perfect power representable as the sum of two Fibonacci numbers is $3864^2 = F_{36} + F_{12}$. In other words, they conjectured that the equation \begin{equation}\tag{$\ast$}\label{eq:abstract} y^a = F_n + F_m \end{equation} has no solutions with $a\geq 2$ and $y^a > 3864^2$. While this is still an open problem, there exist several partial results. For example, recently Kebli, Kihel, Larone and Luca proved an explicit upper bound for $y^a$, which depends on the size of $y$. In this paper, we find an explicit upper bound for $y^a$, which only depends on the Hamming weight of $y$ with respect to the Zeckendorf representation. More specifically, we prove the following: If $y = F_{n_1}+ \dots + F_{n_k}$ and equation \eqref{eq:abstract} is satisfied by $y$ and some non-negative integers $n,m$ and $a\geq 2$, then \[ y^a \leq \exp\left(C{(\varepsilon)} \cdot k^{(3+\varepsilon)k^2} \right). \] Here, $\varepsilon >0$ can be chosen arbitrarily and $C(\varepsilon)$ is an effectively computable constant.

math.NT↗
arXiv2023

Twisted Thue equations with multiple exponents in fixed number fields

Let $K$ be a number field of degree $d\geq 3$ and fix $s$ multiplicatively independent algebraic integers $γ_1, \dots, γ_s \in K^*$ that fulfil some technical requirements, which can be vastly simplified to $\mathbb{Q}$-linearly independence, given Schanuel's conjecture. We then consider the twisted Thue equation \[ \left|N_{K/\mathbb{Q}}\left(X-γ_1^{t_1}\cdotsγ_s^{t_s}Y\right)\right| = 1, \] and prove that it has only finitely many solutions $(x,y, (t_1, \dots, t_s) )$ with $xy \neq 0$ and $\mathbb{Q}\left( γ_1^{t_1}\cdots γ_s^{t_s} \right) = K$, all of which are effectively computable.

math.NT↗
arXiv2023

Thue equations over $\mathbb{C}(T)$: The Complete Solution of a Simple quartic family

In this paper we completely solve a simple quartic family of Thue equations over $\mathbb{C}(T)$. Specifically, we apply the ABC-Theorem to find all solutions $(x,y) \in \mathbb{C}[T] \times \mathbb{C}[T]$ to the set of Thue equations $F_{\lambda}(X,Y) = \xi$, where $\xi \in \mathbb{C}^{\times}$ and \begin{equation*} F_{\lambda}(X,Y):=X^4 -\lambda X^3Y -6 X^2Y^2 + \lambda XY^3 +Y^4, \quad \quad \lambda \in \mathbb{C}[T]/\{\mathbb{C}\} \end{equation*} denotes a family of quartic simple forms.

math.NT↗
arXiv2022

On the Diophantine equation $U_n-b^m = c$

Let $(U_n)_{n\in \mathbb{N}}$ be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants $B$ and $N_0$ such that for any $b,c\in \mathbb{Z}$ with $b> B$ the equation $U_n - b^m = c$ has at most two distinct solutions $(n,m)\in \mathbb{N}^2$ with $n\geq N_0$ and $m\geq 1$. Moreover, we apply our result to the special case of Tribonacci numbers given by $T_1= T_2=1$, $T_3=2$ and $T_{n}=T_{n-1}+T_{n-2}+T_{n-3}$ for $n\geq 4$. By means of the LLL-algorithm and continued fraction reduction we are able to prove $N_0=1.1\cdot 10^{37}$ and $B=e^{438}$. The corresponding reduction algorithm is implemented in Sage.

math.NT↗
arXiv2022

On a variant of Pillai's problem with transcendental numbers

In this paper, we study the asymptotic behaviour of the number of solutions $(m, n)\in \mathbb{N}^2$ to the inequality $ | α^n - β^m | \leq x $ when $x$ tends to infinity. Here $α, β$ are given multiplicatively independent complex numbers with $|α| > 1$ and $|β|>1$.

math.NT↗
arXiv2021

On a family of unit equations over simplest cubic fields

Let $a\in \mathbb{Z}$ and $ρ$ be a root of $f_a(x)=x^3-ax^2-(a+3)x-1$, then the number field $K_a=\mathbb{Q}(ρ)$ is called a simplest cubic field. In this paper we consider the family of unit equations $u_1+u_2=n$ where $u_1,u_2\in \mathbb{Z}[ρ]^*$ and $n\in \mathbb{Z}$. We completely solve the unit equations under the restriction $|n|\leq \max\{1,|a|^{1/3}\}$.

math.NT↗
arXiv2021

On sums of Fibonacci numbers with few binary digits

In this paper we completely solve the Diophantine equation $F_n+F_m=2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$, where $F_k$ denotes the $k$-th Fibonacci number. In addition to complex linear forms in logarithms and the Baker-Davenport reduction method, we use $p$-adic versions of both tools.

math.NT↗