On the pseudo-randomness of subsets related to primitive roots

Many results have been proved on the distribution of the primitive roots. These results reflect certain random type properties of the set G p of the primitive roots modulo p . This fact motivates the question that in what extent behaves G p as a random subset of ℤ p ? First a much more general form of this problem is studied by using the notion of pseudo-randomness of subsets of ℤ n which has been introduced and studied recently by Dartyge and Sárközy. This is followed by the study of the pseudo-randomness of a subset of ℤ p defined by index properties. In both cases it turns out that these subsets possess strong pseudo-random properties (the well-distribution measure and correlation measure of order k are small) but the pseudo-randomness is not perfect: there is a pseudo-random measure (the symmetry measure) which is large.