The Pintz–Steiger–Szemerédi estimate for intersective quadratic polynomials in function fields.

Let q [ t ] be the polynomial ring over the finite field q of q elements. For a natural number N , let N be the set of all polynomials in q [ t ] of degree less than N. Let h be a quadratic polynomial over q [ t ]. Suppose that h is intersective, that is, which satisfies (A − A) ∩ (h ( q [ t ]) ∖ { 0 }) ≠ ∅ for any A ⊆ q [ t ] with limsup N → ∞ | A ∩ N | / q N > 0 , where A − A denotes the difference set of A. Let B ⊆ N. Suppose that (B − B) ∩ h ( q [ t ]) ∖ { 0 } = ∅ and that the characteristic of q is not divisible by 2. It is proved that | B | ≤ C N − c log log log N q N for any 0 < c < 1 / log 3 , where C ≥ 1 is a constant depending only on q , h and c. [ABSTRACT FROM AUTHOR]