The Furstenberg–Sárközy theorem and asymptotic total ergodicity phenomena in modular rings

Abstract

The Furstenberg–Sárközy 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 Poincaré 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):=limN→∞⁡1N∑n=1Nμ(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 P∈Q[n] with deg⁡(P)>1 such that P(Z)⊆Z. For any integer N>1 with lpf(N) sufficiently large, for any subsets A and B of Z/NZ such that |A||B|≥δN2, one has Z/NZ=A+B+S, where S={P(n):1≤n≤N}⊂Z/NZ.