The true complexity of a system of linear equations

TL;DR

This paper investigates the uniformity conditions needed for subsets of finite Abelian groups to contain the expected number of solutions to linear systems, revealing that linear uniformity suffices for certain systems previously thought to require higher uniformity.

Contribution

It demonstrates that quadratic Fourier analysis can show linear uniformity controls solutions to some linear systems, challenging prior assumptions.

Findings

Linear uniformity suffices for certain linear systems

Quadratic Fourier analysis reveals new uniformity-solution relationships

Conjecture on necessary and sufficient conditions for solution counts

Abstract

It is well-known that if a subset A of a finite Abelian group G satisfies a quasirandomness property called uniformity of degree k, then it contains roughly the expected number of arithmetic progressions of length k, that is, the number of progressions one would expect in a random subset of G of the same density as A. One is naturally led to ask which degree of uniformity is required of A in order to control the number of solutions to a general system of linear equations. Using so-called "quadratic Fourier analysis", we show that certain linear systems that were previously thought to require quadratic uniformity are in fact governed by linear uniformity. More generally, we conjecture a necessary and sufficient condition on a linear system L which guarantees that any subset A of F_p^n which is uniform of degree k contains the expected number of solutions to L.