Trigonometric polynomials with frequencies in the set of cubes

We prove that for any $\epsilon >0$ and any trigonometric polynomial $f$ with frequencies in the set $\{n^3:N\le n\le N+N^{2\mathrm{/}3-\epsilon }\}$, one has $$\mathrm{\parallel }f{\mathrm{\parallel }}_4\ll {\epsilon }^{-1\mathrm{/}4}\mathrm{\parallel }f{\mathrm{\parallel }}_2$$ with implied constant being absolute. We also show that the set $\{n^3:N\le n\le N+(0.5N{)}^{1\mathrm{/}2}\}$ is a Sidon set.