Isosceles trapezoids of unit area with vertices in sets of infinite planar measure

TL;DR

This paper proves that any measurable planar set with infinite Lebesgue measure contains the vertices of an isosceles trapezoid of unit area, extending to related shapes like triangles, answering Erdős's longstanding questions.

Contribution

It provides the first affirmative proof that infinite measure sets contain specific geometric configurations, including isosceles trapezoids and triangles, resolving Erdős's open problems.

Findings

Infinite measure sets contain isosceles trapezoids of unit area

Sets also contain isosceles and right-angled triangles of specified areas

Addresses and confirms Erdős's conjectures on geometric configurations in measure theory

Abstract

Paul Erd\H{o}s 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 Erd\H{o}s concerning isosceles triangles and right-angled triangles.