arXiv subjects

Sebastian Heintze

Publications and source records attributed to Sebastian Heintze.

18 recordsLinked to original sources

arXiv2026

On the asymptotic average diameter of blocks of uniformly distributed sequences and related results

This paper was triggered by recent results on the maximal `average distance between consecutive points' of uniformly distributed sequences (u.f.d.s.). Here we address a generalized version of this question, consider pairwise maximal/minimal/total distances in blocks/segments of $ d \geq 2 $ consecutive points of u.f.d.s., and derive sharp upper bounds for all three aggregations. Our main idea of proof consists in, firstly, adding degrees of freedom, secondly, translating the resulting problem to a solvable optimization problem over the compact family of $ d $-stochastic measures, and, thirdly, showing that the obtained bounds are also sharp bounds for the original problem.

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arXiv2025

Decidability of multiplicative matrix equations and related Diophantine problems

Some new decidability results for multiplicative matrix equations over algebraic number fields are established. In particular, special instances of the so-called knapsack problem are considered. The proofs are based on effective methods for Diophantine problems in finitely generated domains as presented in the recent book of Evertse and Gy\"ory. The focus lies on explicit bounds for the size of the solutions in terms of heights as well as on bounds for the number of solutions. This approach also works for systems of symmetric matrices which do not form a semigroup. In the final section some related counting problems are investigated.

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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 $.

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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.

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arXiv2022

On the size of a linear combination of two linear recurrence sequences over function fields

Let $ G_n $ and $ H_m $ be two non-degenerate linear recurrence sequences defined over a function field $ F $ in one variable over $ \mathbb{C} $, and let $ \mu $ be a valuation on $ F $. We prove that under suitable conditions there are effectively computable constants $ c_1 $ and $ C' $ such that the bound \begin{equation*} \mu(G_n - H_m) \leq \mu(G_n) + C' \end{equation*} holds for $ \max \{n,m\} > c_1 $.

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arXiv2022

On the growth of multi-recurrences

In this paper we provide a complete proof for a bound on the growth of multi-recurrences which are defined over a number field. The proven bound was already stated by van der Poorten and Schlickewei forty years ago.

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arXiv2022

Pillai's conjecture for polynomials

In this paper we study the polynomial version of Pillai's conjecture on the exponential Diophantine equation \begin{equation*} p^n - q^m = f. \end{equation*} We prove that for any non-constant polynomial $ f $ there are only finitely many vectors $ (n,m,\mathrm{deg}\ p,\mathrm{deg}\ q) $ with integers $ n,m \geq 2 $ and non-constant polynomials $ p,q $ such that Pillai's equation holds. Moreover, we will give some examples that there can still be infinitely many possibilities for the polynomials $ p,q $.

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arXiv2021

Asymptotics for Pillai's problem with polynomials

Let $ a_1(x)p_1(x)^n + \cdots + a_k(x)p_k(x)^n $ as well as $ b_1(x)q_1(x)^m + \cdots + b_l(x) q_l(x)^m $ be two polynomial power sums where the complex polynomials $ p_i(x) $ and $ q_j(x) $ are all non-constant. Then in the present paper we will give an asymptotic for the number of pairs $ (n,m) \in \mathbb{N}^2 $ such that the degree of the sum of these two power sums is between $ 0 $ and $ d $ when $ d $ goes to infinity.

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arXiv2021

$ S $-unit values of $ G_n + G_m $ in function fields

In this paper we consider a simple linear recurrence sequence $ G_n $ defined over a function field in one variable over the field of complex numbers. We prove an upper bound on the indices $ n $ and $ m $ such that $ G_n + G_m $ is an $ S $-unit. This is a function field analogue of already known results in number fields.

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arXiv2021

Approximation of values of algebraic elements over the ring of power sums

Let $ \mathbb{Q}\mathcal{E}_{\mathbb{Z}} $ be the set of power sums whose characteristic roots belong to $ \mathbb{Z} $ and whose coefficients belong to $ \mathbb{Q} $, i.e. $ G : \mathbb{N} \rightarrow \mathbb{Q} $ satisfies \begin{equation*} G(n) = G_n = b_1 c_1^n + \cdots + b_h c_h^n \end{equation*} with $ c_1,\ldots,c_h \in \mathbb{Z} $ and $ b_1,\ldots,b_h \in \mathbb{Q} $. Furthermore, let $ f \in \mathbb{Q}[x,y] $ be absolutely irreducible and $ \alpha : \mathbb{N} \rightarrow \overline{\mathbb{Q}} $ be a solution $ y $ of $ f(G_n,y) = 0 $, i.e. $ f(G_n,\alpha(n)) = 0 $ identically in $ n $. Then we will prove under suitable assumptions a lower bound, valid for all but finitely many positive integers $ n $, for the approximation error if $ \alpha(n) $ is approximated by rational numbers with bounded denominator. After that we will also consider the case that $ \alpha $ is a solution of \begin{equation*} f(G_n^{(0)}, \ldots, G_n^{(d)},y) = 0, \end{equation*} i.e. defined by using more than one power sum and a polynomial $ f $ satisfying some suitable conditions. This extends results of Bugeaud, Corvaja, Luca, Scremin and Zannier.

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arXiv2020

Diophantine equations in separated variables and polynomial power sums

We consider Diophantine equations of the shape $ f(x) = g(y) $, where the polynomials $ f $ and $ g $ are elements of power sums. Using a finiteness criterion of Bilu and Tichy, we will prove that under suitable assumptions infinitely many rational solutions $ (x,y) $ with a bounded denominator are only possible in trivial cases.

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arXiv2020

Integral zeros of a polynomial with linear recurrences as coefficients

Let $ K $ be a number field, $ S $ a finite set of places of $ K $, and $ \mathcal{O}_S $ be the ring of $ S $-integers. Moreover, let $$ G_n^{(0)} Z^d + \cdots + G_n^{(d-1)} Z + G_n^{(d)} $$ be a polynomial in $ Z $ having simple linear recurrences of integers evaluated at $ n $ as coefficients. Assuming some technical conditions we give a description of the zeros $ (n,z) \in \mathbb{N} \times \mathcal{O}_S $ of the above polynomial. We also give a result in the spirit of Hilbert irreducibility for such polynomials.

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arXiv2020

A function field variant of Pillai's problem

In this paper, we consider a variant of Pillai's problem over function fields $ F $ in one variable over $ \mathbb{C} $. For given simple linear recurrence sequences $ G_n $ and $ H_m $, defined over $ F $ and satisfying some weak conditions, we will prove that the equation $ G_n - H_m = f $ has only finitely many solutions $ (n,m) \in \mathbb{N}^2 $ for any non-zero $ f \in F $, which can be effectively bounded. Furthermore, we prove that under suitable assumptions there are only finitely many effectively computable $ f $ with more than one representation of the form $ G_n - H_m $.

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arXiv2020

A Polynomial Variant of Diophantine Triples in Linear Recurrences

Let $ (G_n)_{n=0}^{\infty} $ be a polynomial power sum, i.e. a simple linear recurrence sequence of complex polynomials with power sum representation $ G_n = f_1α_1^n + \cdots + f_kα_k^n $ and polynomial characteristic roots $ α_1,\ldots,α_k $. For a fixed polynomial $ p $, we consider triples $ (a,b,c) $ of pairwise distinct non-zero polynomials such that $ ab+p, ac+p, bc+p $ are elements of $ (G_n)_{n=0}^{\infty} $. We will prove that under a suitable dominant root condition there are only finitely many such triples if neither $ f_1 $ nor $ f_1 α_1 $ is a perfect square.

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arXiv2020

On the growth of linear recurrences in function fields

Let $ (G_n)_{n=0}^{\infty} $ be a non-degenerate linear recurrence sequence with power sum representation $ G_n = a_1(n) α_1^n + \cdots + a_t(n) α_t^n $. In this paper we will prove a function field analogue of the well known result that in the number field case, under some non-restrictive conditions, for $ n $ large enough the inequality $ \vert G_n\vert \geq \left( \max_{j=1,\ldots,t} \vert α_j\vert \right)^{n(1-\varepsilon)} $ holds true.

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arXiv2020

Norm form equations with solutions taking values in a multi-recurrence

We are interested in solutions of a norm form equation that takes values in a given multi-recurrence. We show that among the solutions there are only finitely many values in each component which lie in the given multi-recurrence unless the recurrence is of precisely described exceptional shape. This gives a variant of the question on arithmetic progressions in the solution set of norm form equations.

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arXiv2019

Perfect powers in polynomial power sums

We prove that a non-degenerate simple linear recurrence sequence $ (G_n(x))_{n=0}^{\infty} $ of polynomials satisfying some further conditions cannot contain arbitrary large powers of polynomials if the order of the sequence is at least two. In other words we will show that for $ m $ large enough there is no polynomial $ h(x) $ of degree $ \geq 2 $ such that $ (h(x))^m $ is an element of $ (G_n(x))_{n=0}^{\infty} $. The bound for $ m $ depends here only on the sequence $ (G_n(x))_{n=0}^{\infty} $. In the binary case we prove even more. We show that then there is a bound $ C $ on the index $ n $ of the sequence $ (G_n(x))_{n=0}^{\infty} $ such that only elements with index $ n \leq C $ can be a proper power.

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arXiv2019

Yet another $S$-unit variant of Diophantine tuples

We show that there are only finitely many triples of integers $ 0 < a < b < c $ such that the product of any two of them is the value of a given polynomial with integer coefficients evaluated at an $ S $-unit that is also a positive integer. The proof is based on a result of Corvaja and Zannier and thus is ultimately a consequence of the Schmidt subspace theorem.

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