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 $\{n^3:N\le n\le N+N^{2/3-\epsilon }\}$, one has $$\Vert f{\Vert }_4\ll {\epsilon }^{-1/4}\Vert f{\Vert }_2$$ with implied constant being absolute. We also show that the set $\{n^3:N\le n\le N+(0.5N{)}^{1/2}\}$ is a Sidon set.

6 pages; inaccuracy in the proof of the main theorem is corrected