Trigonometric Polynomials with Frequencies in the Set of Cubes

We prove that for any $\varepsilon>0$ and any trigonometric polynomial $f$ with frequencies in the set $\{n^3\colon N \leqslant n\leqslant N+N^{2/3-\varepsilon}\}$, one has
\begin{equation*} \|f\|_4 \ll \varepsilon^{-1/4}\|f\|_2 \end{equation*}
with implied constant being absolute. We also show that the set $\{n^3\colon N\leqslant n\leqslant N+(0.5N)^{1/2}\}$ is a Sidon set.