I am wondering if the following result may be true : consider a trigonometric polynomial of "almost periodic" type : $$ f(x)=\sum_{k=0}^n c_k e^{i\lambda_k x} ,\qquad x\in \mathbb{R}$$ We moreover assume that the frequencies $\lambda_k$ are real and spaced as follows $$ \lambda_0\geq 0 \qquad \mbox{and}\qquad \lambda_{k+1}-\lambda_k\geq 1 $$ My question is : does there exist a positive number $T>0$ (a sort of period) and a constant $C>0$ (independent of $f,n$ and the sequence $(\lambda_k)$) such that $$ \sup\limits_{x\in \mathbb{R}}|f(x)|\leq C \sup\limits_{|x|\leq T} |f(x)| $$ Such a question comes from the obvious fact that the answer is positive if each $\lambda_k$ belongs to $a\mathbb{Z}$ for a same $a>0$.
Thanks for having read!