Equivalence of weak quadratic and nilspace characters

Prove that every 1-bounded weak quadratic character on a finite abelian group is close, in the appropriate quantitative sense, to a 2-step nilspace character, thereby establishing an approximate equivalence among weak quadratic characters, quadratic characters, and 2-step nilspace characters.

Background

The paper proves that every 2-step nilspace character of bounded complexity is a quadratic character and hence a weak quadratic character. The converse—recovering a nilspace-character representation from the elementary weak quadratic-character condition—is not established. The authors identify this as a possible route toward a more conceptual equivalence of quadratic-structure notions and potentially more efficient inverse-theorem proofs.