# nt.number theory – Linear independence of algebraic integers of equal norm, part II

In a previous question, I asked whether for a given degree $$n$$ number field $$K$$ whether there exist algebraic integers $$alpha_1, cdots, alpha_n$$, pairwise non-associates and having equal norm exceeding one in absolute value, which are linearly independent. This is answered in three separate answers. One answer (due to Gerry Myerson) addresses the phenomenon that if $$K$$ contains a subfield $$L$$ which is not $$mathbb{Q}$$, then one can sample $$n$$ elements from $$mathcal{O}_L$$ having equal norm which would then of course be $$mathbb{Q}$$-linearly dependent. Noam Elkies gave a general construction (based on an idea of Kenny Lau) which produces for any field $$K$$ with $$n geq 3$$, primitive or not, elements of equal norm which all lie in a single 2-dimensional subspace, which answers the first question linked above in the strongest negative sense.

My question now is the following follow-up: let $$K$$ be a number field with $$(K:mathbb{Q}) = n geq 3$$, and let $$mathcal{O}_K$$ be its ring of integers. Put

$$displaystyle S_K = {a in mathbb{Z}: |a| > 1, exists alpha_1, cdots, alpha_k text{ s.t. }$$
$$displaystyle N_{K/mathbb{Q}}(alpha_j) = a, alpha_j text{ pairwise non-associates and } exists P_a text{ s.t. } alpha_1, cdots, alpha_k},$$

where $$P_a$$ is a rational 2-dimensional subspace of $$K$$. For each $$a in S_K$$ put $$T(a)$$ for the set of planes $$P_a$$ which appear in the definition of $$S_K$$ (that is, all of the planes $$P$$ containing $$n$$ non-associate algebraic integers with norm equal to $$a$$).

Fixing a basis of $$mathcal{O}_K$$, we can identify $$P_a$$ with a subspace of $$mathbb{Q}^n$$, and hence as an element in the Grassmannian $$text{Gr}(2,n)$$. As such each $$P_a$$ can be assigned a height, say $$H$$. My question is this: is it true that

$$displaystyle liminf_{substack{|a| rightarrow infty \ a in S_K \ P_a in T(a)}} H(P_a) = infty?$$

That is, is it true that each plane $$P$$ can only lie in $$T(a)$$ for finitely many $$a in S_K$$?