Trigonometric Polynomials with Frequencies in the Set of Squares and Divisors in a Short Interval

Abstract

Let \(\gamma _0=\frac{\sqrt{5}-1}{2}=0.618\ldots \) . 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 $$\begin{aligned} \Vert f\Vert _4 \ll \varepsilon ^{-1/4}\Vert f\Vert _2 \end{aligned}$$ holds, which makes a progress on a conjecture of Cilleruelo and Córdoba. 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 \([N^{1/2}, N^{1/2}+N^{1/2-\varepsilon }]\) .