Shaffaf 2018 solution erdos ulam problem rational distance

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corollary_1: Shaffaf's corollary, drawn from the proof of Theorem 2 and so resting on the Bombieri-Lang conjecture, that a rational set with infinitely many points not all on a line has all but at most 4 points on a line or all but at most 3 on a circle.

theorem_1: Shaffaf's theorem that for m = 2g + 2 points of the plane with g >= 2, not all on a line, the projectivized distance surface in P^3 is a surface of general type.

theorem_2: Shaffaf's theorem that, assuming the Bombieri-Lang conjecture, there is no dense rational distance subset of the plane.

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Shaffaf, Jafar, A solution of the Erdős-Ulam problem on rational distance sets assuming the Bombieri-Lang conjecture. Discrete Comput. Geom. 60 (2018), no. 2, 283-293. DOI 10.1007/s00454-018-0003-3. The copy read for this card is the arXiv preprint arXiv:1501.00159v3 (20 April 2018), titled "A Proof of the Erdős-Ulam Problem Assuming Bombieri-Lang Conjecture"; its labels and pages are the ones cited here.

Shaffaf attaches to a finite planar set A = {(alpha_i, beta_i)} the 'distance surface' z^2 = product over i of ((x - alpha_i)^2 + (y - beta_i)^2) in P^3, so that points of a rational distance set become rational points of this surface over a suitable number field. Theorem 1 shows that when m = 2g + 2 is even with g >= 2 and the m points are not all collinear, the projectivized distance surface is of general type. Combining this with the Weak Lang (Bombieri-Lang) conjecture, which forbids Zariski-dense rational points on varieties of general type, Theorem 2 concludes that there is no dense rational distance set in the plane, answering the 1945 Erdos-Ulam problem conditionally. Corollary 1, drawn from the proof of Theorem 2 and so also resting on Bombieri-Lang, sharpens this: an infinite rational set that is not contained in a line has all of its points, apart from at most 4, on one line, or all apart from at most 3 on one circle, so the Huff-Peeples elliptic-curve examples are essentially the largest possible. The paper reviews the surrounding results - Anning-Erdos on integral sets, Solymosi-De Zeeuw via Faltings, and the conditional uniform-boundedness input - and the author notes the result is a demonstration of the strength of Bombieri-Lang rather than an unconditional proof. For problem 212 on dense rational-distance sets, Theorem 2 gives a negative answer conditional on the Bombieri-Lang conjecture; it leaves the unconditional problem open.

Source: https://arxiv.org/abs/1501.00159. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1501.00159), every other right reserved.

Read status: claims checked for Theorems 1 and 2, Corollary 1, Lemma 1 (p. 3) and Lemma 2 (p. 4, used in the proof of Theorem 2), read clause by clause on the page images of the arXiv print; the proofs of Theorem 2 and Corollary 1 followed, the proof of Theorem 1 read for structure only. Nothing here is independently reviewed. Result pages: theorem_1, theorem_2 and corollary_1.

Bears on. #212: Theorem 2 (p. 3) answers the problem no, conditional on the Bombieri-Lang conjecture, and Corollary 1 (p. 3), under the same conjecture, puts all but at most 4 points of an infinite rational set not all on a line on one line or all but at most 3 on a circle. The conjecture is unproven; the paper proves nothing unconditional about the problem.

Results.

• Theorem 1 (p. 3): for $m=2g+2$ points of the plane with $g\ge2$, not all on a line, the projectivized distance surface in $\mathbb{P}^3$ is of general type.
• Theorem 2 (p. 3): "Assuming Bombieri-Lang Conjecture, there is no dense rational distance subset in the plane."
• Corollary 1 (p. 3): a rational set with infinitely many points not all on a line has all but at most 4 points on a line or all but at most 3 on a circle; its proof runs through that of Theorem 2, so it too assumes Bombieri-Lang.
• Lemma 1 (p. 3, quoted by the paper from Solymosi-de Zeeuw): inversion centred at a point $x$ of a rational set $S$ with rational radius maps $S\setminus\{x\}$ to a rational set. It has no result page here, since the paper quotes it without proof and uses it nowhere.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.