Simultaneous Polynomial Recurrence

Abstract

Let $A\subseteq \{1,...,N\}$ and $P_1,...,P_{\ell }\in \mathbb{Z}[n]$ with $P_i(0)=0$ and $°P_i=k$ for every $1\le i\le \ell$. We show, using Fourier analytic techniques, that for every $\VE>0$, there necessarily exists $n\in \mathbb{N}$ such that \[\frac{|A\cap (A+P_i(n))|}{N}>(\frac{|A|}{N})^2-\VE\] holds simultaneously for $1\le i\le \ell$ (in other words all of the polynomial shifts of the set $A$ intersect $A$ "$\VE$-optimally"), as long as $N\geq N_1(\VE,P_1,...,P_\ell)$. The quantitative bounds obtained for $N_1$ are explicit but poor; we establish that $N_1$ may be taken to be a constant (depending only on $P_1,...,P_{\ell }$) times a tower of 2's of height $C_{k,\ell}^*+C\eps^{-2}$.

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