Simultaneous Polynomial Recurrence
Neil Lyall, Akos Magyar

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 and with and for every . We show, using Fourier analytic techniques, that for every , there necessarily exists such that \[\frac{|A\cap (A+P_i(n))|}{N}>(\frac{|A|}{N})^2-\VE\] holds simultaneously for (in other words all of the polynomial shifts of the set intersect "-optimally"), as long as . The quantitative bounds obtained for are explicit but poor; we establish that may be taken to be a constant (depending only on ) times a tower of 2's of height .
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