Asymptotic Total Ergodicity and Polynomial Patterns in Finite Fields

A version of the polynomial Szemer´edi theorem was shown to hold in finite fields in [BLM05]. In particular, one has patterns ${x, x + P1(n), . . . , x + Pk(n)}$ (1) for polynomials with zero constant term in large subsets of finite fields. When the polynomials Pi(n) are independent, one can get a substantial refinement regarding the size of subsets in which the patterns (1) can be found. This was done by Peluse ([P19]) for finite fields of sufficiently large characteristic. In joint work with Ethan Ackelsberg ([AB23+]), we establish a complementary result for Fpk when p is fixed and k is large enough. Our result also holds without the assumption that Pi(0) = 0. Our approach is guided by the fact that the dynamics of large fields acting on themselves is asymptotically totally ergodic. The phenomenon of asymptotic total ergodicity is actually quite general and is valid for families of finite rings including $Z/NZ ([BB23]) and Fq[t]/Q(t)Fq[t]$ ([AB23+]) under some natural conditions on $N ∈ N and Q(t) ∈ Fq[t].$ This, in turn, leads to the phenomenon of asymptotic joint ergodicity along independent polynomials. As a consequence, we obtain an asymptotic count (similar to that obatined in [P19]) on the number of patterns in (1) in the case of independent polynomials when the characteristic p is fixed (and q = pk grows). Similar combinatorial results hold asymptotically for more general families of rings ([AB23+, BB23+]). [AB23+] Ethan Ackelsberg and Vitaly Bergelson. Ongoing joint work. [BB23] Vitaly Bergelson and Andrew Best. The Furstenberg-S´ark¨ozy theorem and asymptotic total ergodicity phenomena in modular rings. J. Number Theory, 243 (2023) 615–645. [BB23+] Vitaly Bergelson and Andrew Best. Ongoing joint work. [BLM05] Vitaly Bergelson, Alexander Leibman, and Randall McCutcheon. Polynomial Szemer´edi theorems for countable modules over integral domains and finite fields. J. Anal. Math., 95 (2005) 243–296 [P19] Sarah Peluse. On the polynomial Szemer´edi theorem in finite fields. Duke Math J., 168 (2019) 749–774.