The Furstenberg-S\'ark\"ozy Theorem and Asymptotic Total Ergodicity Phenomena in Modular Rings
Abstract
The Furstenberg-S\'ark\"ozy theorem asserts that the difference set E-E of a subset E ⊂ N with positive upper density intersects the image set of any polynomial P ∈ Z[n] for which P(0)=0. Furstenberg's approach relies on a correspondence principle and a polynomial version of the Poincar\'e recurrence theorem, which is derived from the ergodic-theoretic result that for any measure-preserving system (X,B,μ,T) and set A ∈ B with μ(A) > 0, one has c(A):= N ∞ 1N Σn=1N μ(A T-P(n)A) > 0. The limit c(A) will have its optimal value of μ(A)2 when T is totally ergodic. Motivated by the possibility of new combinatorial applications, we define the notion of asymptotic total ergodicity in the setting of modular rings Z/NZ. We show that a sequence of modular rings Z/NmZ, m ∈ N, is asymptotically totally ergodic if and only if lpf(Nm), the least prime factor of Nm, grows to infinity. From this fact, we derive some combinatorial consequences, for example the following. Fix δ ∈ (0,1] and a (not necessarily intersective) polynomial Q ∈ Q[n] such that Q(Z) ⊂eq Z, and write S = \ Q(n) : n ∈ Z/NZ\. For any integer N > 1 with lpf(N) sufficiently large, if A and B are subsets of Z/NZ such that |A||B| ≥ δ N2, then Z/NZ = A + B + S.