Multivariate Polynomial Values in Difference Sets

Abstract

For $\ell \ge 2$ and $h\in \mathbb{Z}[x_1,\dots ,x_{\ell }]$ of degree $k\ge 2$, we show that every set $A\subseteq \{1,2,\dots ,N\}$ lacking nonzero differences in $h({\mathbb{Z}}^{\ell })$ satisfies $|A|{\ll }_{h}N{e}^{-c(\log N{)}^{\mu }}$, where $c=c(h)>0$, $\mu =[(k-1{)}^2+1{]}^{-1}$ if $\ell =2$, and $\mu =1/2$ if $\ell \ge 3$, provided $h({\mathbb{Z}}^{\ell })$ contains a multiple of every natural number and $h$ satisfies certain nonsingularity conditions. We also explore these conditions in detail, drawing on a variety of tools from algebraic geometry.

46 pages

References in corpus (4)

• Diophantine equations in the primes
• On a theorem of Sárközy for difference sets and shifted primes
• A Maximal Extension of the Best-Known Bounds for the Furstenberg-Sárközy Theorem
• Lower bounds in the polynomial Szemerédi theorem