On sets of polynomials whose difference set contains no squares

Volume 161 / 2013

Thái Hoàng Lê, Yu-Ru Liu Acta Arithmetica 161 (2013), 127-143 MSC: 11P55, 11T55. DOI: 10.4064/aa161-2-2


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


  • Thái Hoàng LêDepartment of Mathematics
    The University of Texas at Austin
    1 University Station, C1200
    Austin, TX 78712, U.S.A.
  • Yu-Ru LiuDepartment of Pure Mathematics
    Faculty of Mathematics
    University of Waterloo
    Waterloo, ON, Canada N2L 3G1

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