The Furstenberg-Sarkoezy theorem for intersective polynomials in function fields

摘要

Let F-q[t] be the polynomial ring over the finite field F-q of q elements. For a natural number N = 2, let G(N) be the set of all polynomials in F-q[t] of degree less than N. Let h(x) is an element of F-q[t][x] be an intersective polynomial, that is, which satisfies (A - A) boolean AND (h(F-q[t]) {0}) not equal empty set for any A subset of F-q[t] [t] with lim sup(N -infinity )vertical bar A boolean AND G(N)/q(N) 0. Let B subset of G(N). Suppose that (B - B) boolean AND (h(F-q[t]){0}) = empty set and 2 = degh p, the characteristic of F-q. It is proved that vertical bar B vertical bar = Cq(N) (log N/N) (1/k-1), where C 1 is a constant depending only on h(x) and q. (C) 2019 Elsevier Inc. All rights reserved.