On the polynomial equivalence of subsets E and f(E) of Z

For a subset E of an integral domain D and an integer-valued polynomial f o ver D, we investigate conditions under which the subsets E and f(E) of D de termine the same integer-valued polynomials on D (this is the definition of polynomial equivalence of E and f(E)). Our primary interest in this proble m lies in the case where D is the ring of rational integers. Using work of McQuillan, the case where E is finite is resolved completely in Section 3. For E infinite we show in several cases that polynomial equivalence of E an d f(E) implies that f is linear, but whether this is true in general for, s ay, D = Z is an open question.