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

Abstract

Let D be the ring of integers of a number field K, P a prime of D for which q = | D / P D | is finite, ν P 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 + P ℓ D 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 { α ( n ) = ν P ( I n ) : n = 1 , 2 , 3 , … } where I n 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 lim n → ∞ α ( n ) / n for any homogeneous set, apply it to the special case of the homogeneous sets Z ∖ P ℓ Z ⊆ 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.