On a conjecture of Erdős over function fields

Abstract Based on Katz’s equidistribution framework, we give a one-dimensional proof of a function-field analogue of a conjecture of Erdős in the large- q regime: for squarefree f of degree n , every class in $$(\mathbb {F}_q[t]/(f))^\times $$ ( F q [ t ] / ( f ) ) × is represented as a product of two monic irreducibles of degrees $$\le n$$ ≤ n once q is sufficiently large in terms of n . The proof yields a uniform $$q^{-1/2}$$ q - 1 / 2 -saving in the relevant twisted character sums, with an explicit constant depending on n . Sawin [12] proved a stronger and more general result by higher-dimensional sheaf-theoretic methods; the present note instead emphasizes a simpler one-dimensional argument within Katz’s framework.