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

The Furstenberg–Sárközy theorem asserts that the difference set E?E of a subset E?N with positive upper density intersects image any polynomial P?Z[n] for which P(0)=0. Furstenberg's approach relies on correspondence principle and version Poincaré recurrence theorem, is derived from ergodic-theoretic result measure-preserving system (X,B,?,T) A?B ?(A)>0, one has c(A):=limN???1N?n=1N?(A?T?P(n)A)>0. limit c(A) will have its optimal value ?(A)2 when T totally ergodic. Motivated by possibility new combinatorial applications, we define notion asymptotic total ergodicity in setting modular rings Z/NZ. We show sequence Z/NmZ,m?N, asymptotically ergodic if only lpf(Nm), least prime factor Nm, grows to infinity. From this fact, derive some consequences, example following. Fix ??(0,1] (not necessarily intersective) P?Q[n] deg?(P)>1 such P(Z)?Z. For integer N>1 lpf(N) sufficiently large, subsets A B Z/NZ |A||B|??N2, Z/NZ=A+B+S, where S={P(n):1?n?N}?Z/NZ.