Sparse mean value estimates, algebraic number solution counting, and non-Archimedean Fourier analysis

We demonstrate two applications of Fourier decoupling theorems over non-Archimedean local fields to real-variable problems. These include short mean value estimates for exponential sums, canonical-scale mean value estimates for exponential sums arising from phase functions with coefficients arising from the traces of powers of algebraic numbers, and solution counting bounds for Vinogradov systems whose indeterminates are families of algebraic numbers. We also record an example where real and $\mathfrak{p}$-adic decoupling estimates differ.