Isosceles trapezoids of unit area with vertices in sets of infinite planar measure
Junnosuke Koizumi
TL;DR
The paper answers Erdős's question by showing that any measurable set of infinite Lebesgue measure in the plane contains the vertices of an isosceles trapezoid of area $1$, and, more generally, that an unbounded set of positive measure contains vertices of an isosceles triangle of area $1$ and of a right-angled triangle of area $1$. The core method uses a Lebesgue density theorem argument together with a rotation-based map $f$ defined by $f(r\cos\theta,r\sin\theta)=(r\cos(\theta+\varphi(r)),r\sin(\theta+\varphi(r)))$ with $\varphi(r)=\arcsin(2/r^2)$ (so $J_f(p)=1$) to realize area-$1$ triangles; a variant $g(p)=(p+f(p))/2$ yields right triangles. For trapezoids, the authors adapt the map via $\psi(r)=\arcsin\left(\frac{R^2}{R^2-1}\cdot\frac{2}{r^2}\right)$ to construct a four-point configuration $p,f(p),R^{-1}f(p),R^{-1}p$ with area $1$, and use a density argument on $S_R=S\cap(R\cdot S)$, with a limiting step ensuring sufficient density. These ingredients together settle Erdős's remaining questions on unit-area trapezoids and related triangles, delineating the role of infinite-measure sets.
Abstract
Paul Erdős posed the question of whether every measurable planar set of infinite Lebesgue measure contains the four vertices of an isosceles trapezoid of unit area. In this paper, we provide an affirmative answer to this question. Additionally, we present affirmative solutions to similar questions by Erdős concerning isosceles triangles and right-angled triangles.