HIGHER UNIFORMITY OF BOUNDED MULTIPLICATIVE FUNCTIONS IN SHORT INTERVALS ON AVERAGE

The paper’s Theorem 1.5 states exactly that if a 1-bounded multiplicative f has large average local Gowers U^{k+1}-norm on intervals [x,x+H] with X^θ ≤ H ≤ X^{1−θ} (0<θ<1/2), then f must be pretentious at the Archimedean scale X^{k+1}/H^{k+1} with bounded conductor, i.e. M(f; C X^{k+1}/H^{k+1}, Q) ≪ 1 . The paper also shows this result is equivalent (via the inverse theorem for Gowers norms) to a nilsequence-correlation statement (Theorem 4.3) with the same X^{k+1}/H^{k+1} scale , and defines the pretentiousness distance M(f;X,Q) in the same way the model uses it . The model’s solution reproduces this structure: (i) pass from an averaged lower bound to many good intervals (the paper similarly discretizes to ≫ X/H intervals) ; (ii) apply the inverse theorem to obtain local correlation with bounded-complexity k-step nilsequences (paper: Theorem 1.5 follows from, and is equivalent to, Theorem 4.3 via the inverse theorem) ; (iii) invoke the short-interval nilsequence orthogonality unless f is pretentious at scale X^{k+1}/H^{k+1} (paper: Theorem 4.3) ; and (iv) contrapose to obtain pretentiousness, matching the paper’s conclusion and scale. Minor presentational differences aside, the model’s argument aligns with the paper’s equivalence-of-statements route and parameter regime, including the scale X^{k+1}/H^{k+1}.