In this paper we revisit the deterministic version of the Sparse Fourier Transform problem, which asks to read only a few entries of $x in mathbb{C}^n$ and design a recovery algorithm such that the output of the algorithm approximates $hat x$, the Discrete Fourier Transform (DFT) of $x$. The randomized case has been well-understood, while the main work in the deterministic case is that of Merhi et al.@ (J Fourier Anal Appl 2018), which obtains $O(k^2 log^{-1}k cdot log^{5.5}n)$ samples and a similar runtime with the $ell_2/ell_1$ guarantee. We focus on the stronger $ell_{infty}/ell_1$ guarantee and the closely related problem of incoherent matrices. We list our contributions as follows. 1. We find a deterministic collection of $O(k^2 log n)$ samples for the $ell_infty/ell_1$ recovery in time $O(nk log^2 n)$, and a deterministic collection of $O(k^2 log^2 n)$ samples for the $ell_infty/ell_1$ sparse recovery in time $O(k^2 log^3n)$. 2. We give new deterministic constructions of incoherent matrices that are row-sampled submatrices of the DFT matrix, via a derandomization of Bernsteins inequality and bounds on exponential sums considered in analytic number theory. Our first construction matches a previous randomized construction of Nelson, Nguyen and Woodruff (RANDOM12), where there was no constraint on the form of the incoherent matrix. Our algorithms are nearly sample-optimal, since a lower bound of $Omega(k^2 + k log n)$ is known, even for the case where the sensing matrix can be arbitrarily designed. A similar lower bound of $Omega(k^2 log n/ log k)$ is known for incoherent matrices.
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