Horizontal character and bump function

Quantitative Leibman theorem states that if a polynomial sequence $g\colon [N]\to G $ fails to be δ-equidistributed, then there is a horizontal character $\psi \colon G/\Gamma \to \mathbb R/\mathbb Z $ of size $\ll \delta ^{-O(1)} $ such that $\Vert \eta \circ g \Vert _{C^\infty [N]} \ll \delta ^{-O(1)} $.

The converse to this statement is trivial but let us write it down explicitly here.