On sets of polynomials whose difference set contains no squares

Thái Hoàng Lê¹ ; Yu-Ru Liu²

¹ Department of Mathematics The University of Texas at Austin 1 University Station, C1200 Austin, TX 78712, U.S.A.
² Department of Pure Mathematics Faculty of Mathematics University of Waterloo Waterloo, ON, Canada N2L 3G1

Acta Arithmetica, Tome 161 (2013) no. 2, pp. 127-143.

Résumé

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

Détail

DOI : 10.4064/aa161-2-2

Keywords: mathbb polynomial ring finite field mathbb mathbb subset mathbb containing polynomials degree strictly define maximal cardinality set subseteq mathbb which a a contains squares polynomials combining polynomial hardy littlewood circle method density increment technology developed pintz steiger szemer prove log