Let $\mathcal{K}=\{-K, -K+1, \cdots ,K-1, K\}\backslash \{0\}$, where $K$ is a positive integer.
Let $\phi(x)=\displaystyle\sum_{k \in \mathcal{K}}e^{\mathrm{i}k x}$, where $\mathrm{i}=\sqrt{-1}$.
I would like to know if the set defined below has zero Lebesgue measure in $\mathbb{R}^N$, where $N > 2K+1$:
$A = \{(x_1, x_2, \cdots, x_N): \phi(x_n) - \phi(x_{n-1}) = x_n - x_{n-1}, \text{ for } n \in \{2, 3, \cdots, N\}, \text{ every } x_n \in (0, 2\pi), \text{ and } x_1 < x_2, \cdots < x_N\}.$