This property was introduced in 2001 by Bombieri and Zannier, who raised the question of whether it holds for fields with uniformly bounded local degrees. They also remarked that, for a (possibly infinite) Galois extension of the rationals whose local degrees are bounded at (at least) one prime, property (N) is implied by the divergence of a certain sum, but suggested that this phenomenon might occur only for number fields. In 2011 Widmer gave a criterion for an infinite extension of the rationals to have property (N) under some condition on the growth of the discriminants of certain finite subextensions of the field.

In this talk I will present several results obtained in this context with A. Fehm. In particular, we show the existence of infinite Galois extensions of the rationals for which the sum considered by Bombieri and Zannier is divergent and to which Widmer's criterion does not apply and we also show the existence of fields without property (N) and having (non-uniformly) bounded local degrees at all primes. This last result is a corollary of a theorem of Fili on totally $S$-adic numbers of small height, of which I will present an effective version.