On sets of polynomials whose difference set contains no squares

Abstract

Let Fq[t] be the polynomial ring over the finite field Fq, and let GN be the subset of Fq[t] containing all polynomials of degree strictly less than N. Define D(N) to be the maximal cardinality of a set A⊆GN for which A−A contains no squares of polynomials. By combining the polynomial Hardy–Littlewood circle method with the density increment technology developed by Pintz, Steiger and Szemerédi, we prove that D(N)≪qN(logN)7/N.