On the square peg problem

We show that if $γ$ is a Jordan curve in $\mathbb{R}^2$ which is close to a $C^2$ Jordan curve $β$ in $\mathbb{R}^2$, then $γ$ contains an inscribed square. In particular, if $κ> 0$ is the maximum unsigned curvature of $β$ and there is a map $f$ from the image of $γ$ to the image of $β$ with $||f(x) - x|| < \frac{1}{10 κ}$ and $f \circ γ$ having winding number $1$, then $γ$ has an inscribed square of positive sidelength.

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