On some applications of finitely generated semi-groups

Abstract

Let V be a finitely generated multiplicative semi-group with r generators in the ring of integers $\mathbb{Z}_\mathbb{K}$of an algebraic number field $\mathbb{K}$of degree n over ℚ. We use various bounds for character sums to obtain results on the distribution of the residues of elements of V modulo an integer ideal q. In the simplest case, when $\mathbb{K} = \mathbb{Q}$and r = 1 this is a classical question on the distribution of residues of an exponential function, which may be interpreted as concerning the quality of the linear congruential pseudo-random number generator. Besides this well known application we consider several other problems from algebraic number theory, the theory of function fields over a finite field, complexity theory, cryptography, and coding theory where results on the distribution of some group V modulo q play a central role.