In this paper we show that, if an increasing sequence $\Lambda=(\lambda_k)_{k\in\Z}$ has gaps going to infinity $\lambda_{k+1}-\lambda_k\to +\infty$ when $k\to\pm\infty$, then for every $T>0$ and every sequence $(a_k)_{k\in\Z}$ and every $N\geq 1$, $$ A\sum_{k=0}^N\frac{|a_k|}{1+k}\leq\frac{1}{T}\int_{-T/2}^{T/2} \abs{\sum_{k=0}^Na_ke^{2i\pi\lambda_k t}}\,\mbox{d}t $$ further, if $\dst\sum_{k\in\Z}\dfrac{1}{1+|\lambda_k|}<+\infty$, $$ B\max_{|k|\leq N}|a_k|\leq\frac{1}{T}\int_{-T/2}^{T/2} \abs{\sum_{k=-N}^Na_ke^{2i\pi\lambda_k t}}\,\mbox{d}t $$ where $A,B$ are constants that depend on $T$ and $\Lambda$ only. The first inequality was obtained by Nazarov for $T>1$ and the second one by Ingham for $T\geq 1$ under the condition that $\lambda_{k+1}-\lambda_k\geq 1$. The main novelty is that if those gaps go to infinity, then $T$ can be taken arbitrarily small. The result is new even when the $\lambda_k$'s are integers where it extends a result of McGehee, Pigno and Smith. The results are then applied to observability of Schr\"odinger equations with moving sensors.