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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.