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 \(\{n^3: N \leq n\leq N+N^{2/3-\varepsilon}\}\) one has \(\|f\|_4 \ll \varepsilon^{-1/4}\|f\|_2\) 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.