Renling Jin

Solution to the inverse problem for upper asymptotic density

Abstract Inverse problems study the structure of a set A when the “size” of A + A is small. In the article, the structure of an infinite set A of natural numbers with positive upper asymptotic density is characterized when A is not a subset of an infinite arithmetic progression of difference greater than one and A + A has the least possible upper asymptotic density. For example, if the upper asymptotic density α of A is strictly between 0 and 1/2, the upper asymptotic density of A + A is equal to 3α/2, and A is not a subset of an infinite arithmetic progression of difference greater than one, then A is either a large subset of the union of two infinite arithmetic progressions with the same common difference k = 2/α or for every increasing sequence hn of positive integers such that the relative density of A in [0, hn ] approaches α, the set A ∩ [0, hn ] can be partitioned into two parts A ∩ [0, cn ] and A ∩ [bn , hn ], such that cn/hn approaches 0, i.e. the size of A ∩ [0, cn ] is asymptotically small compared with the size of [0, hn ], and (hn − bn )/hn approaches α, i.e. the size of A ∩ [bn , hn ] is asymptotically almost the same as the size of the interval [bn , hn ]. The results here answer a question of the author in [R. Jin, Inverse problem for upper asymptotic density, Trans. Amer. Math. Soc. 355 (2003), No. 1, 57–78.]

The isomorphism property versus the special model axiom

This paper answers some questions of D. Ross in RR. In x1, we show that some consequences of the @ 0 or @ 1 special model axiom in RR can not be proved by the isomorphism property for any cardinal. In x2, we show that with one exception, the @ 0 isomorphism property does imply the remaining consequences of the special model axiom in RR. In x3, we improve a result in RR by showing that the special model axiom is equivalent to the @ 0 special model axiom plus saturation.

Applications of Nonstandard Analysis in Additive Number Theory

This paper reports recent progress in applying nonstandard analysis to additive number theory, especially to problems involving upper Banach density.

High density piecewise syndeticity of sumsets

Abstract Renling Jin proved that if A and B are two subsets of the natural numbers with positive Banach density, then A + B is piecewise syndetic. In this paper, we prove that, under various assumptions on positive lower or upper densities of A and B , there is a high density set of witnesses to the piecewise syndeticity of A + B . Most of the results are shown to hold more generally for subsets of Z d . The key technical tool is a Lebesgue density theorem for measure spaces induced by cuts in the nonstandard integers.

The strength of the isomorphism property

In x1 of this paper, we characterize the isomorphism property of nonstandard universes in terms of the realization of some second{order types in model theory. In x2, several applications are given. One of the applications answers a question of D. Ross in [R] about inflnite Loeb measure spaces.

Inverse problem for upper asymptotic density

For a set A of natural numbers, the structural properties are described when the upper asymptotic density of 2A + {0,1} achieves the infimum of the upper asymptotic densities of all sets of the form 28 + {0, 1}, where the upper asymptotic density of B is greater than or equal to the upper asymptotic density of A. As a corollary, we prove that if the upper asymptotic density of A is less than 1 and the upper asymptotic density of 2A + {0,1} achieves the infimum, then the lower asymptotic density of A must be 0.

Better Nonstandard Universes with Applications

There are various reasons why some nonstandard universes are considered to be better than others. Different people may have different opinions and may choose different nonstandard universes to work within for different purposes. We think it is reasonable to consider that a nonstandard universe which possesses stronger power for deriving results in either standard or nonstandard mathematics, or which supplies more convenient tools so that, in practice, some complicated derivation procedures admit significant simplifications, is better than the one which doesn’t.

Existence of Some Sparse Sets of Nonstandard Natural Numbers

Answers are given to two questions concerning the existence of some sparse subsets of ℋ = {0, 1 … ., H – 1} ⊆ *ℕ. where H is a hyperfinite integer. In §1. we answer a question of Kanovei by showing that for a given cut U in ℋ , there exists a countably determined set X ⊆ ℋ which contains exactly one element in each U -monad, if and only if U = a · ℕ for some a Є ℋ ∖ {0}. In §2, we deal with a question of Keisler and Leth in [6]. We show that there is a cut V ⊆ ℋ such that for any cut U , (i) there exists a U -discrete set X ⊆ ℋ with X + X = ℋ ( mod H ) provided , (ii) there does not exist any U -discrete set X ⊆ ℋ with X + X = ℋ ( mod H ) provided . We obtain some partial results for the case U = V .

Maharam spectra of Loeb spaces

We characterize Maharam spectra of Loeb probability spaces and give some applications of the results.

Possible size of an ultrapower of \(\omega\)

Abstract. Let