OBJECTIVE AND SOURCE REVIEW
Erdős #212; independent attempt, 2026-09-23. Can a rational-distance set be dense in the plane? I read all of
212.tex: its quadratic-field normalization, genus-two line obstruction and inversion argument. Here is an elementary obstruction to constructing such a set as an additive group, requiring no algebraic-geometry theorem.
WORK: ADDITIVE CLOSURE IS IMPOSSIBLE
An additive subgroup $G\subset\mathbb R^2$ whose elements all have rational norm lies on a line. Suppose otherwise that $u,v\in G$ are independent. Write $a=|u|\in\mathbb Q_{>0}$, $b=u\cdot v\in\mathbb Q$, $c=|v|^2\in\mathbb Q$. Rationality of $b$ follows by subtracting $|u+v|^2-|u|^2-|v|^2$. Choose a positive integer $K$ such that $A=Ka$, $B=Kb/a$, and $K^2c$ are integers. Then, for every integer $m\ge1$, \[
(K|mu+v|)^2=(Am+B)^2+D,\qquad
D=K^2c-B^2>0.
\] Here $D$ is a positive integer, by independence and Cauchy--Schwarz. The left side is a rational square and an integer, so it must be the square of an integer (reduce a rational square root to lowest terms). For sufficiently large $m$, however, \[
(Am+B)^2<(Am+B)^2+D<(Am+B+1)^2,
\] an impossibility. This proves the claim.
ADVERSARIAL VERIFICATION AND REMAINING GAP
Only closure under $mu+v$ is used; general rational-distance sets have no such closure. The integer scaling is chosen once, independent of $m$. The positivity $D>0$ fails exactly in the collinear case, where rational additive groups can indeed be dense on a line. The lemma rules out dense additive groups and their translates as constructions, but gives no obstruction to irregular countable sets lacking additive closure. No extension theorem or unrestricted nonexistence result is obtained.
SOURCES CHECKED
Complete local source:
gpt_pro_5.4/212.tex. Official indexed statement checked at
https://www.erdosproblems.com/212, 2026-09-23. The subgroup proof above is self-contained and is not claimed to be new in the literature.
FINAL
UNRESOLVED. Additive subgroup constructions are excluded; arbitrary dense sets remain untreated.
COMPLETION ESTIMATE
Conservative subjective progress toward planar density.
COMPLETION: 10%