Optimal Extrapolation Bounds for Sparse Fourier Sums

We prove a sharp extrapolation bound for $k$-sparse Fourier sums with arbitrary real frequencies and no separation assumption. For every $g(t)=\sum_{j=1}^k v_je^{iλ_jt}$ with $λ_j\in\mathbb{R}$ and every $x\geq 1$, \begin{align*} |g(x)|\le k\exp(O(k\operatorname{arcosh} x))\|g\|_{L^2[-1,1]}. \end{