On sums of S-integers of bounded norm

  title={On sums of S-integers of bounded norm},
  author={Christopher Frei and Robert F. Tichy and Volker Ziegler},
  journal={Monatshefte f{\"u}r Mathematik},
We prove an asymptotic formula for the number of $$S$$S-integers in a number field $$K$$K that can be represented by a sum of $$n$$n$$S$$S-integers of bounded norm. 
Constrained Triangulations, Volumes of Polytopes, and Unit Equations
An equivalent and easy-to-check combinatorial criterion for the facets of $\mathcal{P}$ is proved and this leads to an exact formula for the volume of a polytope arising in the theory of unit equations.
30 years of collaboration
This paper focuses on two topics in more detail, 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 originated from a question of Zelinsky ( on the unit sum number problem).


On quantitative aspects of the unit sum number problem
We investigate the function uK,S(n; q) which counts the number of representations of algebraic integers α with $${|N_{K/{\mathbb Q}}(\alpha)| \leq q}$$ for some real positive q that can be written as
On Sums of Units
Abstract.It is shown that if R is a finitely generated integral domain of zero characteristic, then for every n there exist elements of R which are not sums of at most n units. This applies in
Counting the values taken by algebraic exponential polynomials
We prove an effective mean-value theorem for the values of a nondegenerate, algebraic exponential polynomial in several variables. These objects generalise simultaneously the fundamental examples of
Linear forms with algebraic coefficients. I
Abstract In a recent paper I generalized Roth's well-known theorem on rational approximation to an algebraic number α in two dual directions. Namely, I proved a theorem on simultaneous rational
Arithmetic Progressions in Linear Combinations of S-Units
M. Pohst asked the following question: is it true that every prime can be written in the form 2u ± 3v with some non-negative integers u, v? We put the problem into a general framework, and prove that
The number of families of solutions of decomposable form equations
In [16], Schmidt introduced the notion of family of solutions of norm form equations and showed that there are only finitely many such families. In [18], Voutier gave an explicit upper bound for the
Linear equations in variables which Lie in a multiplicative group
Let K be a field of characteristic 0 and let n be a natural number. Let r be a subgroup of the multiplicative group (K*) n of finite rank r. Given a 1 ,...,a n E K* write A(a 1 ,...,a n , Γ) for the
On the quantitative unit sum number problem—an application of the subspace theorem
lie in finitely many proper subspaces of Q. As an application of the subspace theorem Schmidt [16] described all norm form equations that have finitely many solutions. The subspace theorem has been
Linearformen mit algebraischen Koeffizienten
In a recent paper I generalized the Thue-Siegel-Roth-Schmidt theorem on simultaneous rational approximation of n real algebraic numbers to include also the p-adic case. In the present paper I shall
Additive unit representations in rings over global fields - A survey
We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. We focus on rings of integers in number fields and in