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Source. Theorem 2.1, p. 2, of Jozsef Solymosi and Frank de Zeeuw, On a question of Erdős and Ulam, arXiv:0806.3095v2 (14 January 2009), published in Discrete Comput. Geom. 43 (2010), no. 2, 393-401, the version named on the source card; the proof is Section 3, pp. 2-6.

Read depth. Claims checked: the statement and the definition of a rational set (p. 1) were read clause by clause on the printed pages. The proof was read for structure only. Nothing here is independently reviewed.

Statement

Setting (pp. 1-2). A rational set is a set $S\subset\mathbb R^2$ in which the distance between any two elements is a rational number. A curve over a field $K\subset\mathbb R$ is the zero set in $\mathbb R^2$ of a polynomial in two variables with coefficients in $K$; its genus is that of the projective variety the polynomial defines (p. 2).

Theorem 2.1 (p. 2, quoted). "Every rational set of the plane has only finitely many points in common with an algebraic curve defined over $\mathbb R$, unless the curve has a component which is a line or a circle."

Equivalently, an irreducible real algebraic curve containing an infinite rational set is a line or a circle; the paper presents this as proving, for algebraic curves, Erdős' conjecture that a set with a dense rational subset should be very special (abstract). Lines and circles are genuine exceptions: every line contains a dense rational set, and so does the unit circle (p. 1).

Proof pointer

The main tool is Faltings' theorem: a curve of genus at least 2 defined over a number field has only finitely many rational points (p. 2). By Lemma 3.4 (p. 3), after a similarity taking two points of $S$ to $(0,0)$ and $(1,0)$, every point of $S$ has the form $(r_1,r_2\sqrt k)$ with $r_1,r_2\in\mathbb Q$ for a square-free integer $k$ depending on $S$, so a curve holding enough points of $S$ is defined over $\mathbb Q(\sqrt k)$; this settles curves of genus at least 2 (p. 2). For an irreducible curve of genus 1 (Section 3.5, pp. 3-4), and of genus 0 and degree at least 4 (Section 3.6, p. 4), the points of $S$ lift to points of the space curve cut out by the curve and the cone $x^2+y^2=z^2$, whose genus the Riemann-Hurwitz formula shows to be at least 2 after a suitable rotation. For genus 0 and degree 2 or 3 other than a line or a circle (Section 3.7, pp. 4-6), inversion centred at the origin, which is birational, takes a conic to a cubic and raises some cubics to degree 5, which the earlier case covers; each remaining cubic is moved so that its singularity is at the origin, and a parametrization by lines through the origin turns infinitely many points of $S$ into infinitely many solutions of a hyperelliptic equation of degree 8 and genus 3.

Dependencies

Faltings' theorem (cited, p. 2); Lemma 3.3 (inversion preserves rational sets, p. 3); Lemma 3.4 (the almost-rational form of a rational set, p. 3, attributed by the paper to Kemnitz); the Riemann-Hurwitz formula.

Bears on

• Problem 212: a dense subset of the plane has infinitely many points off every line and every circle. With Theorem 2.2 the theorem shows that a dense rational set, if one exists, meets every real algebraic curve in only finitely many points; the paper does not draw this consequence out and does not settle the problem.