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On a Conjecture of Erdős over Function Fields

Likun Xie

TL;DR

This paper proves a function-field analogue of Erdős’ conjecture in the large-$q$ regime using Katz's one-dimensional convolution–equidistribution framework. By constructing a weight-0 middle-extension sheaf and pushing it to a base torus, it obtains a $q^{-1/2}$ saving for twists by totally ramified characters, yielding an error $O(n^{n} q^{n-1/2})$ and establishing $E(k,f)=B^{\times}$ for all squarefree $f$ of degree $n$ once $q$ is large enough, with $Q(n)$ effectively bounded by $n^{\kappa n}$ (e.g., $\kappa=23$). This provides a simpler, dimension-one alternative to Sawin’s higher-dimensional square-root cancellation while remaining robust across variants of the multiplicative structure (e.g., different representations of the Dirichlet coefficients). The results extend to related multiplicative functions and degree-splitting scenarios, illustrating the versatility of Katz’s framework for function-field arithmetic questions in the large-$q$ limit.

Abstract

Using Katz's equidistribution framework, we show that for any squarefree polynomial $f \in \mathbb{F}_q[t]$ of degree $n \ge 2$, every residue class modulo $f$ can be represented as a product of two monic irreducible polynomials of degree at most $n$, provided $q$ is sufficiently large in terms of $n$. This gives the function-field analogue of a conjecture of Erdős in the large-$q$ regime. Sawin previously proved this representation with stronger square-root cancellation via a higher-dimensional sheaf-theoretic construction. This note presents a one-dimensional argument that yields a natural $q^{-1/2}$ saving.

On a Conjecture of Erdős over Function Fields

TL;DR

This paper proves a function-field analogue of Erdős’ conjecture in the large- regime using Katz's one-dimensional convolution–equidistribution framework. By constructing a weight-0 middle-extension sheaf and pushing it to a base torus, it obtains a saving for twists by totally ramified characters, yielding an error and establishing for all squarefree of degree once is large enough, with effectively bounded by (e.g., ). This provides a simpler, dimension-one alternative to Sawin’s higher-dimensional square-root cancellation while remaining robust across variants of the multiplicative structure (e.g., different representations of the Dirichlet coefficients). The results extend to related multiplicative functions and degree-splitting scenarios, illustrating the versatility of Katz’s framework for function-field arithmetic questions in the large- limit.

Abstract

Using Katz's equidistribution framework, we show that for any squarefree polynomial of degree , every residue class modulo can be represented as a product of two monic irreducible polynomials of degree at most , provided is sufficiently large in terms of . This gives the function-field analogue of a conjecture of Erdős in the large- regime. Sawin previously proved this representation with stronger square-root cancellation via a higher-dimensional sheaf-theoretic construction. This note presents a one-dimensional argument that yields a natural saving.
Paper Structure (3 sections, 5 theorems, 102 equations)

This paper contains 3 sections, 5 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Initial set-up
  3. Bound for $R(a,n)$

Key Result

Theorem 1.1

Let $n \ge 2$. Then there exists a constant $Q(n) > 0$ such that for every finite field $k = \mathbb{F} _q$ with $q \ge Q(n)$ and every squarefree polynomial $f \in k[X]$ of degree $n$, we have Moreover, one may take $Q(n)$ to be of the form $n^{\,\kappa n}$ for some absolute constant $\kappa > 0$; for instance, $\kappa = 23$ suffices.

Theorems & Definitions (12)

  • Theorem 1.1
  • Definition 2.1
  • Definition 3.1
  • Remark 3.1
  • Theorem 3.2: katz
  • Proposition 3.3
  • proof
  • Definition 3.2
  • Proposition 3.4
  • proof
  • ...and 2 more