Trigonometric polynomials with frequencies in the set of squares

Abstract

Let ${\gamma }_0=\frac{\sqrt{5}-1}{2}=0.618\dots$ . We prove that, for any $\epsilon >0$ and any trigonometric polynomial $f$ with frequencies in the set $\{n^2:N⩽n⩽N+N^{{\gamma }_0-\epsilon }\}$, the inequality $$\Vert f{\Vert }_4\ll {\epsilon }^{-1/4}\Vert f{\Vert }_2$$ 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 $\epsilon >0$, there is $C(\epsilon )>0$ such that each positive integer $N$ has at most $C(\epsilon )$ divisors in the interval $[N^{1/2},N^{1/2}+N^{1/2-\epsilon }]$