Koizumi 2025 isosceles trapezoids unit area vertices sets

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theorem_1: Koizumi's theorem that every unbounded measurable subset of the plane of positive Lebesgue measure contains the three vertices of an isosceles triangle of area 1 and the three vertices of a right-angled triangle of area 1.

theorem_2: Koizumi's theorem that every measurable subset of the plane of infinite Lebesgue measure contains the four vertices of an isosceles trapezoid of area 1, with the paper's stated counterexample for unbounded sets of positive measure.

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Junnosuke Koizumi, Isosceles trapezoids of unit area with vertices in sets of infinite planar measure. arXiv:2501.01914 (2025). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2501.01914), every other right reserved. The paper appeared in Proc. Amer. Math. Soc., published online 2025-08-29, DOI 10.1090/proc/17322 (Crossref); the copy read for this card is arXiv v1 (3 January 2025; the print is dated January 6, 2025), whose pages the results below cite.

Koizumi answers affirmatively three of the five Erdős questions from the 1983 Oberwolfach Measure Theory proceedings about unit-area polygons with vertices in planar sets of infinite measure. Theorem 1 is stronger than asked: it needs only an unbounded measurable set of positive Lebesgue measure, and finds in it both an isosceles triangle and a right-angled triangle of area 1, each with all three vertices in the set. Theorem 2 finds, in any measurable set of infinite Lebesgue measure, an isosceles trapezoid of area 1 with all four vertices in the set, and the author notes this fails for merely unbounded sets of positive measure (counterexample: a small disk together with the points (n,0) for n = 1, 2, 3, ...). The method, inspired by Kovac-Predojevic, builds an area-preserving rotation-by-phi(r) diffeomorphism f with phi(r)=arcsin(2/r^2), so that O, p, f(p) always span a unit-area isosceles triangle, and then uses Lebesgue's density theorem to force both p and f(p) into the set; for right triangles f(p) is replaced by (p+f(p))/2, and trapezoids come from truncating the apex. The paper records (p. 1) that Kovač and Predojević had answered the cyclic-quadrilateral question yes and the congruent-sides question no, and that the trapezoid and the two triangle questions appeared unresolved, citing Problem #353 of erdosproblems.com.

Source: https://arxiv.org/abs/2501.01914.

Read status. Claims checked: Theorems 1 and 2 and the remark after Theorem 2 were read clause by clause on the printed pages, and the proofs (pp. 2-5) were followed. Nothing here is independently reviewed.

Bears on. #353: of the problem's five questions about measurable planar sets of infinite measure, Theorem 2 answers the isosceles-trapezoid question yes, and Theorem 1, which needs only an unbounded set of positive measure, answers the isosceles-triangle and right-angled-triangle questions yes. The paper does not treat the cyclic quadrilateral or the convex polygon with congruent sides.

Results.

• Theorem 1 (p. 1): every unbounded measurable planar set of positive Lebesgue measure contains the vertices of an isosceles triangle of area 1 and of a right-angled triangle of area 1.
• Theorem 2 (p. 2): every measurable planar set of infinite Lebesgue measure contains the four vertices of an isosceles trapezoid of area 1; the remark after it (p. 2) states that unbounded sets of positive measure need not, giving the disk x^2+y^2 <= 1/100 together with the points (n,0), n = 1, 2, 3, ..., as a counterexample.

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