In this paper we show that, if an increasing sequence \(\Lambda =(\lambda _k)_{k\in {\mathbb {Z}}}\) has gaps going to infinity \(\lambda _{k+1}-\lambda _k\rightarrow +\infty \) when \(k\rightarrow \pm \infty \) , then for every \(T>0\) and every sequence \((a_k)_{k\in {\mathbb {Z}}}\) and every \(N\ge 1\) , \(\begin{aligned} A\sum _{k=0}^N\frac{|a_k|}{1+k}\le \frac{1}{T}\int _{-T/2}^{T/2} {\left| {\sum _{k=0}^Na_ke^{2i\pi \lambda _k t}}\right| }\,\text{ d }t \end{aligned}\) further, if \(\displaystyle \sum _{k\in {\mathbb {Z}}}\dfrac{1}{1+|\lambda _k|}<+\infty \) , \(\begin{aligned} B\max _{|k|\le N}|a_k|\le \frac{1}{T}\int _{-T/2}^{T/2} {\left| {\sum _{k=-N}^Na_ke^{2i\pi \lambda _k t}}\right| }\,\text{ d }t \end{aligned}\) 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 \(T\ge 1\) under the condition that \(\lambda _{k+1}-\lambda _k\ge 1\) . 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 Schrödinger equations with moving sensors.
