Banded quadratic digit functions along irreducible polynomials over finite fields

Abstract

Let $q$ be an odd prime power and let $\F_q$ be the finite field with $q$ elements. Let $\mathcal{P}(n)$ be the set of monic irreducible polynomials of degree $n$ over $\mathbb{F}_q$. For $f=t^n+f_{n-1}t^{n-1}+\cdots+f_0\in\mathcal{P}(n)$, fix coefficients $c_0,\ldots,c_m\in\mathbb{F}_q$ with $c_m\ne0$ and put $$ Q_A(f)=\sum_{j=0}^m c_j\sum_{i=j}^n f_i f_{i-j}+\ell_n(f),$$ where $\ell_n$ is an arbitrary linear form in the coefficients of $f$ and $f_n=1$. We prove that $Q_A$ is equidistributed on $\mathcal{P}(n)$: for every $\gamma\in\mathbb{F}_q$, $$\#\{f\in\mathcal{P}(n):Q_A(f)=\gamma\}=\frac{\#\mathcal{P}(n)}{q}+O_A(q^{19n/20+o(n)}),$$ as \(n\to\infty\), with $q$ and the quadratic band fixed. This extends the finite-field Rudin--Shapiro result from nearest-neighbour correlations to arbitrary fixed symmetric Laurent symbols. The proof combines Vaughan's identity with rank estimates for Toeplitz forms; the main new ingredient is an averaged rank-defect estimate for reciprocal symbols in the central Type I range.