A quantitative version of Tao's result on the Toeplitz Square Peg Problem

TL;DR

This paper extends Tao's work by demonstrating that specific planar curves, formed by two 1-Lipschitz functions, necessarily contain inscribed squares with side lengths proportionally bounded below, providing a quantitative enhancement of the Square Peg Problem.

Contribution

It offers a quantitative version of Tao's result, establishing a lower bound on inscribed square size for curves composed of two 1-Lipschitz functions.

Findings

Inscribed squares exist with side length proportional to the maximum difference of the functions.

Provides a universal constant lower bound for the inscribed square's side length.

Extends the understanding of the Square Peg Problem for specific classes of curves.

Abstract

Building on a result by Tao, we show that a certain type of simple closed curve in the plane given by the union of the graphs of two $1$-Lipschitz functions inscribes a square whose sidelength is bounded from below by a universal constant times the maximum of the difference of the two functions.