Abstract

Fix a prime power q, and let F_q[t] denote the ring of polynomials over the finite field F_q. Suppose A ⊆ { f ∈ F_q[t] : deg f ≤ N } contains no pair of elements whose difference is of the form P − 1 with P irreducible. Adapting Green’s approach to Sarkozy’s theorem for shifted primes in ℤ using the van der Corput property, we show that

|A| ≪ q^((N+1)(11/12 + o(1))),

improving upon the bound O(q^((1 − c / log N)(N+1))) due to Le and Spencer.

(Joint work with Steve Fan)