Trigonometric polynomials with frequencies in the set of squares

Abstract

Let $\gamma_0=\frac{\sqrt5-1}{2}=0.618\ldots$ . We prove that, for any $\varepsilon>0$ and any trigonometric polynomial $f$ with frequencies in the set $\{n^2: N \leqslant n\leqslant N+N^{\gamma_0-\varepsilon}\}$, the inequality $$ \|f\|_4 \ll \varepsilon^{-1/4}\|f\|_2 $$ holds, which makes a progress on a conjecture of Cilleruelo and Cordoba. We also present a connection between this conjecture and the conjecture of Ruzsa which asserts that, for any $\varepsilon>0$, there is $C(\varepsilon)>0$ such that each positive integer $N$ has at most $C(\varepsilon)$ divisors in the interval $[N^{1/2}, N^{1/2}+N^{1/2-\varepsilon}]$