On the distribution of M-tuples of B-numbers

In the classical sense, the set B consists of all integers which can be written as a sum of two perfect squares. In other words, these are the values attained by norms of integral ideals over the Gaussian field Q(i). G.J. Rieger (1965) and T. Cochrane / R.E. Dressler (1987) established bounds for the number of pairs (n,n+h), resp., triples (n,n+1,n+2) of B-numbers up to a large real parameter x. The present article generalizes these investigations into two directions: The result obtained deals with arbitrary M-tuples of arithmetic progressions of positive integers, excluding the trivial case that one of them is a constant multiple of some other one. Furthermore, the estimate applies to the case of an arbitrary normal extension K of the rational field instead of Q(i).