arXiv subjects

Lajos Hajdu

Publications and source records attributed to Lajos Hajdu.

17 recordsLinked to original sources

arXiv2026

Asymptotic formulas for sums of elements from a multiplicative group

Let $K$ be a number field, $k\geq 2$ an integer, $(K^*)^k$ the $k$-fold direct product of $K^*$ with coordinatewise multiplication, and $Γ$ a finitely generated subgroup of rank $r$ of $(K^*)^k$. Further, let $H(α)$ denote the absolute exponential height of an algebraic number $α$. Fix non-zero elements $a_1,\ldots , a_k\in K$. We give asymptotic formulas for the number of $\mathbf{x}=(x_1,\ldots , x_k)\inΓ$ with $H(a_1x_1+\cdots +a_kx_k)\leq X$ as $X\to\infty$ such that no non-empty subsum of $a_1x_1+\cdots +a_kx_k$ vanishes. By the same method of proof, we obtain an asymptotic formula as $X\to\infty$ for the number of non-negative integers $n$ with $H(u_n)\leq X$, where $\{ u_n\}$ is a linear recurrence sequence.

math.NT↗

arXiv2025

Multiplicative dependence in the denominators of points of elliptic curves

Let $E_1, \ldots, E_s $ be $s$, not necessary distinct, elliptic curves over $\mathbb{Q}$. We give upper bounds on the frequency of $s$-tuples of points in $E_1(\mathbb{Q})\times \ldots \times E_s(\mathbb{Q})$ whose denominators or $x$-coordinates are multiplicatively dependent. More precisely, we give such bounds in two scenarios: one in which we fix $s$ non-torsion $\mathbb{Q}$-rational points $P_i \in E_i(\mathbb{Q})$ and arbitrary $\mathbb{Q}$-rational points $Q_i \in E_i(\mathbb{Q})$, $i =1, \ldots, s$, and we count $s$-tuples \[ (n_1P_1+Q_1,\ldots, n_sP_s+Q_s) \in E_1(\mathbb{Q}) \times \ldots \times E_s(\mathbb{Q}) \] with $n_1, \ldots, n_s$ in an arbitrary interval of length $N$, and the second in which we count points $(P_1,\ldots,P_s) \in E_1(\mathbb{Q}) \times \ldots \times E_s(\mathbb{Q})$ of bounded canonical height.

math.NT↗

arXiv2025

Multiplicative dependence in linear recurrence sequences

For a wide class of integer linear recurrence sequences $\left(u(n)\right)_{n=1}^\infty$, we give an upper bound on the number of $s$-tuples $\left(n_1, \ldots, n_s\right) \in \left(\mathbb Z\cap [M+1,M+ N]\right)^s$ such that the corresponding elements $u(n_1), \ldots, u(n_s)$ in the sequence are multiplicatively dependent.

math.NT↗

arXiv2024

On Diophantine graphs

Diophantine tuples are of ancient and modern interest, with a huge literature. In this paper we study Diophantine graphs, i.e., finite graphs whose vertices are distinct positive integers, and two vertices are linked by an edge if and only if their product increased by one is a square. We provide various results for Diophantine graphs, including extendability properties, lower- and upper bounds for the maximum number of edges and chromatic numbers.

math.NT↗

arXiv2018

On the smallest number of terms of vanishing sums of units in number fields

Let $K$ be a number field. In the terminology of Nagell a unit $\varepsilon$ of $K$ is called {\it exceptional} if $1-\varepsilon$ is also a unit. The existence of such a unit is equivalent to the fact that the unit equation $\varepsilon_1+\varepsilon_2+\varepsilon_3=0$ is solvable in units $\varepsilon_1,\varepsilon_2,\varepsilon_3$ of $K$. Numerous number fields have exceptional units. They have been investigated by many authors, and they have important applications. In this paper we deal with a generalization of exceptional units. We are interested in the smallest integer $k$ with $k\geq 3$, denoted by $\ell(K)$, such that the unit equation $\varepsilon_1+\dots+\varepsilon_k=0$ is solvable in units $\varepsilon_1,\dots,\varepsilon_k$ of $K$. If no such $k$ exists, we set $\ell(K)=\infty$. Apart from trivial cases when $\ell(K)=\infty$, we give an explicit upper bound for $\ell(K)$. We obtain several results for $\ell(K)$ in number fields of degree at most $4$, cyclotomic fields and general number fields of given degree. We prove various properties of $\ell(K)$, including its magnitude, parity as well as the cardinality of number fields $K$ with given degree and given odd resp. even value $\ell(K)$. Finally, as an application, we deal with certain arithmetic graphs, namely we consider the representability of cycles. We conclude the paper by listing some problems and open questions.

math.NT↗

arXiv2017

On arithmetic progressions in Lucas sequences

In this paper, we consider arithmetic progressions contained in Lucas sequences of first and second kind. We prove that for almost all sequences, there are only finitely many and their number can be effectively bounded. We also show that there are only a few sequences which contain infinitely many and one can explicitly list both the sequences and the progressions in them. A more precise statement is given for sequences with dominant root.

math.NT↗

arXiv2016

On the distribution of polynomials with bounded height

We provide an asymptotic expression for the probability that a randomly chosen polynomial with given degree, having integral coefficients bounded by some B, has a prescribed signature. We also give certain related formulas and numerical results along this line. Our theorems are closely related to earlier results of Akiyama and Pethő, and also yield extensions of recent results of Dubickas and Sha.

math.NT↗

arXiv2016

Perfect powers in products of terms of elliptic divisibility sequences

Diophantine problems involving recurrence sequences have a long history and is an actively studied topic within number theory. In this paper, we connect to the field by considering the equation \begin{align*} B_mB_{m+d}\dots B_{m+(k-1)d}=y^\ell \end{align*} in positive integers $m,d,k,y$ with $\gcd(m,d)=1$ and $k\geq 2$, where $\ell\geq 2$ is a fixed integer and $B=(B_n)_{n=1}^\infty$ is an elliptic divisibility sequence, an important class of non-linear recurrences. We prove that the above equation admits only finitely many solutions. In fact, we present an algorithm to find all possible solutions, provided that the set of $\ell$-th powers in $B$ is given. (Note that this set is known to be finite.) We illustrate our method by an example.

math.NT↗

arXiv2015

30 years of collaboration

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem), which will be presented in turn. This paper evolved from a plenary invited talk that the authors gave at the Joint Austrian-Hungarian Mathematical Conference 2015, August 25-27, 2015 in Győr (Hungary).

math.NT↗

arXiv2014

On conjectures and problems of Ruzsa concerning difference graphs of S-units

Given a finite nonempty set of primes S, we build a graph $\mathcal{G}$ with vertex set $\mathbb{Q}$ by connecting x and y if the prime divisors of both the numerator and denominator of x-y are from S. In this paper we resolve two conjectures posed by Ruzsa concerning the possible sizes of induced nondegenerate cycles of $\mathcal{G}$, and also a problem of Ruzsa concerning the existence of subgraphs of $\mathcal{G}$ which are not induced subgraphs.

math.NT↗

arXiv2012

Bounds for approximate discrete tomography solutions

In earlier papers we have developed an algebraic theory of discrete tomography. In those papers the structure of the functions $f: A \to \{0,1\}$ and $f: A \to \mathbb{Z}$ having given line sums in certain directions have been analyzed. Here $A$ was a block in $\mathbb{Z}^n$ with sides parallel to the axes. In the present paper we assume that there is noise in the measurements and (only) that $A$ is an arbitrary or convex finite set in $\mathbb{Z}^n$. We derive generalizations of earlier results. Furthermore we apply a method of Beck and Fiala to obtain results of he following type: if the line sums in $k$ directions of a function $h: A \to [0,1]$ are known, then there exists a function $f: A \to \{0,1\}$ such that its line sums differ by at most $k$ from the corresponding line sums of $h$.

math.CO↗

arXiv2012

Bounds for discrete tomography solutions

We consider the reconstruction of a function on a finite subset of $\mathbb{Z}^2$ if the line sums in certain directions are prescribed. The real solutions form a linear manifold, its integer solutions a grid. First we provide an explicit expression for the projection vector from the origin onto the linear solution manifold in the case of only row and column sums of a finite subset of $\mathbf{Z}^2$. Next we present a method to estimate the maximal distance between two binary solutions. Subsequently we deduce an upper bound for the distance from any given real solution to the nearest integer solution. This enables us to estimate the stability of solutions. Finally we generalize the first mentioned result to the torus case and to the continuous case.

math.CO↗

arXiv2012

Representing integers as linear combinations of power products

Let P be a finite set of at least two prime numbers, and A the set of positive integers that are products of powers of primes from P. Let F(k) denote the smallest positive integer which cannot be presented as sum of less than k terms of A. In a recent paper Nathanson asked to determine the properties of the function F(k), in particular to estimate its growth rate. In this paper we derive several results on F(k) and on the related function which denotes the smallest positive integer which cannot be presented as sum of less than k terms from the union of A and -A.

math.NT↗

arXiv2012

Approximations of the Euclidean distance by chamfer distances

Chamfer distances play an important role in the theory of distance transforms. Though the determination of the exact Euclidean distance transform is also a well investigated area, the classical chamfering method based upon "small" neighborhoods still outperforms it e.g. in terms of computation time. In this paper we determine the best possible maximum relative error of chamfer distances under various boundary conditions. In each case some best approximating sequences are explicitly given. Further, because of possible practical interest, we give all best approximating sequences in case of small (i.e. 5 by 5 and 7 by 7) neighborhoods.

cs.IT↗

arXiv2011

Representing integers as linear combinations of powers

At a conference in Debrecen in October 2010 Nathanson announced some results concerning the arithmetic diameters of certain sets. He proposed some related results on the representation of integers by sums or differences of powers of 2 and 3. In this note we prove some results on this problem and the more general problem about the representation by linear combinations of powers of some fixed integers.

math.NT↗

arXiv2009

Optimal systems of fundamental S-units for LLL-reduction

We show that a particular parameter plays a vital role in the resolution of S-unit equations, at the stage where LLL-reduction is applied. We define the notion of optimal system of fundamental S-units (with respect to this parameter), and prove that such a system exists and can be effectively constructed. Applying our results and methods, one can obtain much better bounds for the solutions of S-unit equations after the reduction step, than earlier. We briefly also discuss some effects of our results on the method of Wildanger and Smart for the resolution of S-unit equations.

math.NT↗

arXiv2007

Arithmetic progressions of squares, cubes and $n$-th powers

In this paper we continue the investigations about unlike powers in arithmetic progression. We provide sharp upper bounds for the length of primitive non-constant arithmetic progressions consisting of squares/cubes and $n$-th powers.

math.NT↗