Simultaneous Polynomial Recurrence

TL;DR

This paper proves that for polynomial shifts of a set within a large enough finite set, the intersection with the original set can be made arbitrarily close to the expected value, using Fourier analysis.

Contribution

It establishes a polynomial recurrence result with explicit bounds, showing simultaneous near-optimal intersections for multiple polynomial shifts.

Findings

Existence of an n such that all polynomial shifts intersect A nearly optimally.

Explicit bounds for the size of the set needed, depending on polynomial degrees and parameters.

Fourier analytic techniques are used to achieve the recurrence result.

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 $C_{k,\ell}^*+C\eps^{-2}$.