Lemma 5 — unit pairs from a relative-norm fibre

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Statement

Let $F$ be totally real of degree $d$, let $K/F$ be CM with conjugation $c$, let $I$ be a fractional ideal of $K$, and let $0\neq\alpha\in N_{K/F}(I)$. Put

$$M=\#\{\beta\in I:\beta c(\beta)=\alpha\}.$$

For every $R>1$, there is a finite $U\subset\mathbb R^2$ such that

$$|U|\leq \left( 2R\#(N_{K/F}(I)/(\alpha))^{1/(2d)}+1 \right)^{2d} \tag{1}$$

and

$$\frac{D_{\mathrm{ord}}(U)}{|U|} \geq\left(1-\frac1R\right)^{2d}M. \tag{2}$$

Here $D_{\mathrm{ord}}(U)$ counts ordered pairs in $U^2$ at Euclidean distance one.

Proof

Use the norm and injective projection in [[discrete_geometry/sawin_2026_explicit_lower_bound_unit_distance_problem/lemma_4|Lemma 4]]. Its lower bound for the least nonzero lattice norm is

$$\rho\geq\#(N_{K/F}(I)/(\alpha))^{-1/(2d)},$$

and each of the $M$ elements in the norm fiber has lattice norm and projected Euclidean norm equal to one. Inserting these facts into [[discrete_geometry/sawin_2026_explicit_lower_bound_unit_distance_problem/lemma_2|Lemma 2]] gives (1)--(2).

Source scope

This is Lemma 5 on physical p. 5 of the arXiv v1 manuscript.

Used by. [[discrete_geometry/sawin_2026_explicit_lower_bound_unit_distance_problem/proposition_10|Proposition 10]].

Bears on. Problem 90.