Johnson, Keith Limits of characteristic sequences of integer-valued polynomials on homogeneous sets. (English) Zbl 1233.11030 J. Number Theory 129, No. 12, 2933-2942 (2009). Summary: Let \(D\) be the ring of integers of a number field \(K, P\) a prime of \(D\) for which \(q=|D/PD|\) is finite, \(\nu _P\) the corresponding valuation of \(K\) and \(E\) a homogeneous subset of \(D\) with respect to \(P\), i.e. a set with the property \(E=E+P^\ell D\) for some positive integer \(\ell \). Also let \(\text{Int}(E,D)\) 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 \(\{\alpha (n)= \nu_P(I_n): n = 1,2,3,\dots\}\) where \(I_n\) is the fractional ideal formed by 0 and the leading coefficients of elements of \(\text{Int}(E,D)\) of degree \(\leq n\). In this paper we give a recursive method for computing the limit \(\lim_{n\rightarrow \infty}\alpha (n)/n\) for any homogeneous set, apply it to the special case of the homogeneous sets \(\mathbb Z\setminus P^\ell \mathbb Z \subseteq \mathbb Z \) for \(\ell =1,2,3,\dots \) , and show that in general the possible values of this limit as \(E\) ranges over all possible homogeneous subsets are dense in the interval \((1/(q-1),\infty )\). We also apply this method to certain infinite unions of homogeneous sets and obtain formulas for these limits as regular continued fractions. Cited in 11 Documents MSC: 11C08 Polynomials in number theory 13F20 Polynomial rings and ideals; rings of integer-valued polynomials × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Bhargava, M., \(P\)-orderings and polynomial functions on arbitrary subsets of Dedekind rings, J. Reine Angew. Math., 490, 101-127 (1997) · Zbl 0899.13022 [2] Bhargava, M., The factorial function and generalizations, Amer. Math. Monthly, 107, 783-799 (2000) · Zbl 0987.05003 [3] Boulanger, J.; Chabert, J.-L., Asymptotic behavior of characteristic sequences of integer-valued polynomials, J. Number Theory, 80, 238-259 (2000) · Zbl 0973.11093 [4] Boulanger, J.; Chabert, J.-L.; Evrard, S.; Gerboud, G., The characteristic sequence of integer-valued polynomials on a subset, Lect. Notes Pure Appl. Math., 205, 161-174 (1999) · Zbl 0964.13011 [5] Cahen, P.-J.; Chabert, J.-L., Integer Valued Polynomials (1997), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0884.13010 [6] Fekete, M., Uber die verteilung der wurzeln bei gewissen algebraischen gleichungen mit ganzzahligen koeffizienten, Math. Z., 17, 228-249 (1923) · JFM 49.0047.01 [7] K. Johnson, \(P\); K. Johnson, \(P\) · Zbl 1180.13024 [8] Pólya, G.; Szegö, G., Problems and Theorems in Analysis, vol. 1 (1972), Springer-Verlag: Springer-Verlag Berlin · Zbl 0236.00003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.