Trigonometric polynomials with frequencies in the set of cubes

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