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Simultaneous Polynomial Recurrence (1009.0766v1)

Published 3 Sep 2010 in math.CA, math.CO, and math.NT

Abstract: Let $A\subseteq{1,...,N}$ and $P_1,...,P_\ell\in\Z[n]$ with $P_i(0)=0$ and $\deg P_i=k$ for every $1\leq i\leq\ell$. We show, using Fourier analytic techniques, that for every $\VE>0$, there necessarily exists $n\in\N$ such that [\frac{|A\cap (A+P_i(n))|}{N}>(\frac{|A|}{N})2-\VE] holds simultaneously for $1\leq i\leq \ell$ (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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