---

Claim. Terence Tao's blog post The Erdos-Ulam problem, varieties of general type, and the Bombieri-Lang conjecture (20 December 2014) shows that, if the Bombieri-Lang conjecture holds for surfaces over the rationals, no subset of $\mathbb{R}^2$ with all pairwise distances rational is dense, which would answer Problem 212 no. The post supposes $S$ dense with rational distances and normalizes it by a translation, a rotation, a rational dilation and a reflection so that it contains $(0,0)$ and $(1,0)$; a third point then has the form $(a,\sqrt b)$ with $a$ and $b>0$ rational, and every point of $S$ lies in $\{(x,y\sqrt b):x,y\in\mathbb{Q}\}$. Four points $(x_j,y_j\sqrt b)$ of $S$ in general position define the affine surface $V$ of solutions $(x,y,r_1,\ldots,r_4)$ of $(x-x_j)^2+b(y-y_j)^2=r_j^2$ for $j=1,\ldots,4$, whose rational points include the lifts of the points of $S$ with their four distances. Claim 3 of the post asserts that the rational points of $V$ are not Zariski dense; since a dense $S$ would give a Zariski-dense set of them, $S$ cannot be dense. Claim 3 follows from Theorem 4, that the blowup of the projective closure of $V$, a complete intersection of four quadrics in $\mathbb{P}^6$, is a smooth surface of general type, together with the Bombieri-Lang conjecture for that surface. Remark 5 of the post adds that the same argument shows, under the conjecture, that a rational distance set is never Zariski dense, so by the theorem of Solymosi and de Zeeuw all but finitely many of its points lie on one line or one circle. The post records that an unpublished work of Shaffaf had obtained a similar result; Shaffaf's paper, posted eleven days later and refereed, is recorded on its own page.

Hypothesis. The post needs only the Bombieri-Lang conjecture for smooth projective irreducible surfaces of general type defined over $\mathbb{Q}$: the rational points of such a surface are not Zariski dense. The conjecture is unproven, so the claim is conditional and derives no standing for the problem by itself.

Standing. Claimed. The result is a blog post, not refereed; the site's commentary credits it with the conditional answer, but the site labels the problem OPEN, so that credit is commentary and not acceptance, and no evidence is listed.

Depends on. Nothing in this wiki.