Möbius function is strongly orthogonal to polynomial phases over $\mathbb{F}_p[t]$

TL;DR

This work proves power-saving orthogonality between the Möbius function on F_p[t] and polynomial phases of degree k when p > k. The authors reduce correlations to the bias of a symmetric multilinear form L_Q, then leverage a novel approximation for biased multilinear forms and a strong structure theorem for low-codimension biased varieties to force a reduction to lower-degree polynomials, which contradicts the inductive hypothesis. Central contributions include a bias-based approximation theorem for multilinear forms, a bounded-codimension-variety framework for biased forms, and polynomial bounds in the inverse theorem for uniformity norms for polynomial phases in finite vector spaces. The results extend function-field analogues of Möbius orthogonality and provide robust, quantitative control over correlations with polynomial phases over prime fields.

Abstract

In this paper, we prove power-saving bounds for the corelation of the Möbius function with polynomial phases of degree $k$ in function fields $\mathbb{F}_p[t]$, when $p > k$. The proof relies on a new approximation result for phases of biased multilinear forms and the recently established strong bounds for the problem of finding bounded codimension varieties inside the dense ones. Along the way, we also obtain polynomial bounds in the inverse theorem for Gowers uniformity norms in the special case of polynomial phases in finite vector spaces.

Theorems & Definitions (47)

• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 4
• Lemma 5: Bounded codimension implies density
• proof
• Lemma 6
• proof
• Lemma 7
• proof