A subset $E\subset \mathbb Z$ is call $\Lambda(4)$ set, if it has a const $C=C(E)<\infty$ st. for all sequence $a\in l^2(E)$, $\|\sum_{n\in E}a_ne^{inx}\|_{L^4}\le C\|a\|_{l^2}$.
Denote $r_2(E,n)=\#\{(j,k)\in E^2:j+k=n\}$, easy to see if set $E$ satisfy $\sup\limits_{n\in\mathbb Z}r_2(E,n)<\infty$, then $E$ is $\Lambda(4)$.
But conversely, does the condition $\Lambda(4)$ imply $\sup\limits_{n\in\mathbb Z}r_2(E,n)<\infty$?
I read Rudin’s article, , he said “But no counterexample is known”. Is there any progress?