What global properties of a function can we infer from local information? Bernstein-type discretization inequalities offer one answer: they show the supremum norm of a polynomial can be controlled by its absolute supremum on a small, discrete subset of the domain. While such inequalities enjoy widespread use in analysis and approximation theory, their multivariate versions are often limited by a strong dependence on dimension or the need for very many test points. In this talk we show how to get a dimension-free discretization with few points, leading to sharpenings of several inequalities in harmonic analysis over cyclic groups. Along the way we develop a probabilistic technique for iterating one-dimensional inequalities without paying a dimension-dependent price. Based on joint work with Lars Becker, Ohad Klein, Alexander Volberg, and Haonan Zhang.