On sets of polynomials whose difference set contains no squares
Volume 161 / 2013
Acta Arithmetica 161 (2013), 127-143
MSC: 11P55, 11T55.
DOI: 10.4064/aa161-2-2
Abstract
Let ${\mathbb F}_q[t]$ be the polynomial ring over the finite field ${\mathbb F}_q$, and let ${\mathbb G_{N}}$ be the subset of ${\mathbb F}_q[t]$ containing all polynomials of degree strictly less than $N$. Define $D(N)$ to be the maximal cardinality of a set $A \subseteq {\mathbb G_{N}}$ 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 $D(N) \ll q^N(\log N)^{7}/N$.
