Let $\mathrm{\gamma} : [0,1] \to \mathbb{R}^2$ be a piecewise smooth, simple plane curve.
Assume $\gamma(0) = (0,0)$, $\gamma(1) = (1,0)$ and that the slope of the tangent is not $0$ wherever it's defined.
The chord set of $\gamma$ is defined as
$ \{ \delta \ge 0 | \ \exists t_1, t_2: \ \gamma(t_2) - \gamma(t_1) = (\delta,0) \}$.
What can be said of the chord set? I'm particularly interested in its accumulation points.

In the case of $\gamma$ being the graph of a continuous function (having the same value on 0 and 1), the chord set has been studied:

• Levy(1934) showed that the set $\{ \frac{1}{n}, n \in \mathbb{N}\}$ is the only chord set common to all the continuous functions (so-called universal chord theorem).

• For any $\delta \in ]0,1[$, and $f$ continuous function, either $\delta$ is a chord, or $f$ has two distinct chords of length $1-\delta$ (this should have an accessible proof).

Is there any known result in the more general case of (the range of) $\gamma$ not being the graph of a function?

Let $\mathrm{\gamma} : [0,1] \to \mathbb{R}^2$ be a piecewise smooth, simple plane curve.
Assume $\gamma(0) = (0,0)$, $\gamma(1) = (1,0)$ and that the slope of the tangent is not $0$ wherever it's defined.
The chord set of $\gamma$ is defined as
$ \{ \delta \ge 0 | \ \exists t_1, t_2: \ \gamma(t_2) - \gamma(t_1) = (\delta,0) \}$.
What can be said of the chord set? I'm particularly interested in its accumulation points.

In the case of $\gamma$ being the graph of a continuous function (having the same value on 0 and 1), the chord set has been studied:

• Levy(1934) showed that the set $\{ \frac{1}{n}, n \in \mathbb{N}\}$ is the maximal chord set common to all the continuous functions (so-called universal chord theorem).

• For any $\delta \in ]0,1[$, and $f$ continuous function, either $\delta$ is a chord, or $f$ has two distinct chords of length $1-\delta$ (this should have an accessible proof).

Is there any known result in the more general case of (the range of) $\gamma$ not being the graph of a function?

Let $\mathrm{\gamma} : [0,1] \to \mathbb{R}^2$ be a piecewise smooth, simple plane curve.
Assume $\gamma(0) = (0,0)$, $\gamma(1) = (1,0)$ and that the slope of the tangent is not $0$ wherever it's defined.
The chord set of $\gamma$ is defined as
$ \{ \delta \ge 0 | \ \exists t_1, t_2: \ \gamma(t_2) - \gamma(t_1) = (\delta,0) \}$.
What can be said of the chord set? I'm particularly interested in its accumulation points.

In the case of $\gamma$ being the graph of a continuous function (having the same value on 0 and 1), the chord set has been studied:

• Levy(1934) showed that the set $\{ \frac{1}{n}, n \in \mathbb{N}\}$ is the only chord set common to all the continuous functions (so-called universal chord theorem).

• For any $\delta \in ]0,1[$, and $f$ continuous function, either $\delta$ is a chord, or $f$ has two distinct chords of length $1-a$ (this should have an accessible proof).

Is there any known result in the more general case of (the range of) $\gamma$ not being the graph of a function?

Let $\mathrm{\gamma} : [0,1] \to \mathbb{R}^2$ be a piecewise smooth, simple plane curve.
Assume $\gamma(0) = (0,0)$, $\gamma(1) = (1,0)$ and that the slope of the tangent is not $0$ wherever it's defined.
The chord set of $\gamma$ is defined as
$ \{ \delta \ge 0 | \ \exists t_1, t_2: \ \gamma(t_2) - \gamma(t_1) = (\delta,0) \}$.
What can be said of the chord set? I'm particularly interested in its accumulation points.

In the case of $\gamma$ being the graph of a continuous function (having the same value on 0 and 1), the chord set has been studied:

• Levy(1934) showed that the set $\{ \frac{1}{n}, n \in \mathbb{N}\}$ is the only chord set common to all the continuous functions (so-called universal chord theorem).

• For any $\delta \in ]0,1[$, and $f$ continuous function, either $\delta$ is a chord, or $f$ has two distinct chords of length $1-\delta$ (this should have an accessible proof).

Is there any known result in the more general case of (the range of) $\gamma$ not being the graph of a function?