Isosceles trapezoids of unit area with vertices in sets of infinite planar measure (2501.01914v1)

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.

• The paper proves that any measurable set of infinite Lebesgue measure contains isosceles trapezoids of unit area, resolving a longstanding Erdős question.
• The study employs transformation mappings and Lebesgue’s density theorem to systematically locate polygonal vertices within infinite planar sets.
• Findings extend previous results on triangle configurations and offer significant theoretical insights for computational geometry and geometric measure theory.

Analytical Exploration of Polygons with Infinite Planar Measure

The paper authored by Junnosuke Koizumi addresses a question proposed by Paul Erdős concerning geometric configurations within measurable planar sets of infinite Lebesgue measure. Specifically, the focus is directed towards determining the presence of specific polygonal vertices within these sets. Erdős' inquiry involved the feasibility of identifying the four vertices of an isosceles trapezoid of unit area within any measurable planar set of infinite Lebesgue measure, alongside similar geometric configurations in the form of isosceles and right-angled triangles.

Results and Contributions

Koizumi's research presents affirmative solutions to these questions, offering rigorous proofs for the existence of such configurations within the described sets. The paper presents the following main results:

1. Theorem on Isosceles and Right-Angled Triangles: It is established that any unbounded measurable set of positive Lebesgue measure in $\mathbb{R}^2$ contains the vertices of an isosceles triangle of area 1, as well as a right-angled triangle of the same area. This result follows a similar method inspired by the work of Kovač and Predojević, who tackled questions pertaining to different polygonal configurations.
2. Theorem on Isosceles Trapezoids: Extending the theorem to isosceles trapezoids, Koizumi demonstrates that any measurable set of infinite Lebesgue measure includes the four vertices necessary to form isosceles trapezoids of unit area. This finding is significant as it leaves the question resolved for isosceles trapezoids where previous work only addressed other geometric configurations, providing a comprehensive view of Erdős' posed problems.

Methodological Approach

Koizumi employs a sophisticated approach that leverages the properties of transformation mappings and analytical geometry. By defining a function that alters the angular position of points within the plane, the paper demonstrates how such points can be manipulated to form the desired polygons while still residing within the measurable set. This is substantiated using principles from Lebesgue’s density theorem, ensuring that transformed points remain within the confines of the initial set.

Implications and Future Work

These findings have both theoretical and practical implications in the field of geometric measure theory. Theoretically, they solidify an understanding of polygonal structure within expansive measurable spaces, thereby broadening the scope of possibilities for embedding geometric configurations in spaces of infinite measure. Practically, this can influence computational geometry and related fields where understanding the distribution of geometric configurations is essential.

Future research might explore extensions of these results to higher dimensional spaces or other polygonal structures. Additionally, the methods employed could be adapted or enhanced to address more complex questions in geometric measure theory, potentially increasing the scope of Erdős-type problems that can be resolved using such methodologies.

The paper stands as a strong mathematical contribution, expanding the boundaries of known capabilities within measure theory and providing clarity on longstanding questions related to polygonal embeddings in measurable planar sets.