Limits of characteristic sequences of integer-valued polynomials on homogeneous sets

Let D be the ring of integers of a number field K, P a prime of D for which q=|D/PD| is finite, νP the corresponding valuation of K and E a homogeneous subset of D with respect to P, i.e. a set with the property E=E+PℓD for some positive integer ℓ. Also let Int(E,D) denote the ring of polynomials in K[x] which take values in D when evaluated at points of E. The characteristic sequence of E with respect to P is the sequence of integers where In is the fractional ideal formed by 0 and the leading coefficients of elements of Int(E,D) of degree ⩽n. In this paper we give a recursive method for computing the limit limn→∞α(n)/n for any homogeneous set, apply it to the special case of the homogeneous sets Z∖PℓZ⊆Z for ℓ=1,2,3,… , and show that in general the possible values of this limit as E ranges over all possible homogeneous subsets are dense in the interval (1/(q−1),∞). We also apply this method to certain infinite unions of homogeneous sets and obtain formulas for these limits as regular continued fractions.