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By use of a nonstandard model, sequences of natural numbers are associated with a collection of closed subsets of reals in a natural way. The topological and measure-theoretic properties of the associated closed sets are used to prove standard theorems and define new density functions on sequences. 1. The main goal of this work is to continue to provide a framework for the use of nonstandard methods in combinatorial number theory which was begun in [Le]. Whereas only countable models were used in that paper we will concentrate here on those properties which require the use of a larger model, i.e. some degree of saturation or a large language. By results of Friedman [K] the use of countable models cannot lead to theorems whose standard proofs are set-theoretically more complicated than the nonstandard proofs, while Henson, Kaufmann and Keisler [HKK] show that standard proofs of results obtained using some saturation may require correspondingly more choice. In light of this it is perhaps not surprising that the standard theorems generated here seem considerably more difficult to prove in a classical way than those in [Le]. In ?2 for each standard sequence A and infinite number z (i.e. z E *NN) we define "st,(A)" to be the subset of [0,1] obtained by dividing each element of *A by z and taking standard parts. Standard conditions on a sequence A are given which are equivalent to "ste(A) always contains an interval" and "ste(A) never contains an interval." Using techniques similar to those used in [Le] some standard theorems involving these conditions are proved. In ?3 two new density functions for sequences are introduced by using the Lebesgue measure of the sets stz(A). Various properties of these density functions are developed, including some similarities to the classical asymptotic and Schnirelmann densities. The density functions suggest a notion of a "partial basis" which is somewhat analogous to the classical notion of basis. In ?4 various sequence versions of the classical result of Steinhaus that the sum of two sets of real numbers of positive measure contains an interval are developed. Finally, in ?5, some problems and concluding remarks are given. 2. We wish to apply the measure theory of the real numbers to sequences by using a nonstandard model. In order to do so we will need a model which includes a symbol for each real number as well as for each sequence of real numbers, all functions from R' into R, and the symbol "E". We could use a nonstandard Received by the editors July 7, 1986 and, in revised form, May 14, 1987. This paper contains some of the results of the author's doctoral thesis (The University of Colorado, 1985). 1980 Mathemati'cs Subject Classification (1985 Revision). Primary liU10, 03H15, 11B05, 11B75. (?)1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

Applications of nonstandard models and Lebesgue measure to sequences of natural numbers