Linear configuration count

A linear configuration count is a weighted sum over tuples satisfying prescribed linear equations. Orthogonality of complex exponentials often rewrites such a count as an integral of Fourier transforms.

Suppose a linear configuration count has a Fourier-integral representation with $k$ factors. If one has a uniform Fourier bound for the difference between two weights and compatible $L^{k-1}$ bounds for each weight, expanding the difference one factor at a time and applying Hölder's inequality bounds the change in the count.