# Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

## On sets of polynomials whose difference set contains no squares

### Tom 161 / 2013

Acta Arithmetica 161 (2013), 127-143 MSC: 11P55, 11T55. DOI: 10.4064/aa161-2-2

#### Streszczenie

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 $D(N) \ll q^N(\log N)^{7}/N$.

#### Autorzy

• Thái Hoàng LêDepartment of Mathematics
The University of Texas at Austin
1 University Station, C1200
Austin, TX 78712, U.S.A.
e-mail
• Yu-Ru LiuDepartment of Pure Mathematics
Faculty of Mathematics
University of Waterloo