In this paper, we construct Fourier quasicrystals with unit masses in arbitrary dimensions. This generalizes a one-dimensional construction of Kurasov and Sarnak. To do this, we employ a class of complex algebraic varieties avoiding certain regions in Cn, which generalize hypersurfaces defined by Lee–Yang polynomials. We show that these are Delone almost periodic sets that have at most finite intersection with every discrete periodic set.