Steven C. Leth

Descriptive set theory over hyperfinite sets

The separation, uniformization, and other properties of the Borel and projective hierarchies over hyperfinite sets are investigated and compared to the corresponding properties in classical descriptive set theory. The techniques used in this investigation also provide some results about countably determined sets and functions, as well as an improvement of an earlier theorem of Kunen and Miller.

Sequences in countable nonstandard models of the natural numbers

Two different equivalence relations on countable nonstandard models of the natural numbers are considered. Properties of a standard sequence A are correlated with topological properties of the equivalence classes of the transfer of A. This provides a method for translating results from analysis into theorems about sequences of natural numbers.

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.

Some nonstandard methods in combinatorial number theory

A combinatorial result about internal subsets of *N is proved using the Lebesgue Density Theorem. This result is then used to prove a standard theorem about difference sets of natural numbers which provides a partial answer to a question posed by Erdös and Graham.

Near arithmetic progressions in sparse sets

Nonstandard methods are used to obtain results in combinatorial number theory. The main technique is to use the standard part map to translate density properties of subsets of *N into Lebesgue measure properties on [0,1]. This allows us to obtain a simple condition on a standard sequence A that guarantees the existence of intervals in arithmetic progression, all of which contain elements of A with various uniform density conditions.

An asymptotic formula for powers of binomial coefficients

Introduction The importance of teaching our students at all levels how to use computers as a problem-solving tool is well-recognised. Using numerical approximations to discover unknown formulas can be a powerful example. It is usually pointed out to the students that computer exploration alone is not sufficient and should be followed up whenever possible with a proof or at least some additional analysis. The importance of this is illustrated in the example below, in which there is compelling numerical evidence for an incorrect conclusion.

A monad measure space for logarithmic density

We provide a framework for proofs of structural theorems about sets with positive Banach logarithmic density. For example, we prove that if

Meager sets on the hyperfinite time line

A\subseteq \mathbb {N}