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 further developed by Schlickewei [11, 12] and is proved in it’s most general form by Evertse and Schlickewei [3] (see also [13]). These investigations led to many applications, e.g. to the finiteness of the numbers of solutions to S-unit equations (see e.g. [4]) or to estimates for the number of zeros of linear recurrence sequences (see e.g. [18]). In this paper we use these techniques to obtain results on a quantitative version of the so called unit sum number problem. In particular, we solve a problem related to a recent paper of M. Jarden and W. Narkiewicz [9]. The investigation of the unit sum number of rings goes back to the 1950’s, when Zelinsky [20] proved that every element of the endomorphism ring E of a vector space V over a division ring D can be written as the sum of two automorphisms (units in E) unless D is the field with two elements and the dimension of V is one. Following Goldsmith, Pabst and Scott [6] the unit sum number of a ring (with identity) is defined as the samllest number k such that every r ∈ R can be represented as sum of k units. If no such k exists and R is additively generated by