A general method is developed by using nonstandard analysis for formulating and proving a theorem about upper Banach density parallel to each theorem about Shnirel'man density or lower asymptotic density.
Abstract. Answering a problem posed by Keisler and Leth, we prove a theorem in non-standard analysis to reveal a phenomenon about sumsets, which says that if two sets A and B are large in terms of "measure", then the sum A+B is not small in terms of "order-topology". The theorem has several corollaries about sumset phenomenon in the standard world; these are described in sections 2-4. One of these is a new result in additive number theory; it says that if two sets A and B of non-negative integers have positive upper or upper Banach density, then A + B is piecewise syndetic.
A theorem in non-standard analysisLet * V be a non-standard extension of a standard universe V , which contains all standard real numbers. The reader may consult [7] V , we write c > U if c is greater than every element in U . Given a hyperfinite integer H, we always write H for the set {0, 1, . . . , H − 1}. If U is a cut and U ⊆ H, then U is a "small" subset of H because U is closed under addition, hence for every n ∈ N, U < H/n.U -topology and U -nowhere dense are introduced in [6]. A real number R in * V is finite if |R| < n for some standard integer n. Each finite R has a standard part, denoted by st(R), and defined to be the unique standard real number r such that R is infinitesimally close to r. For any internal set A we write |A| for the internal cardinality of A and * P(A) for the set of all internal subsets of A. When a is a standard or non-standard real number, we write |a| for the absolute value of a.
Abstract. Erdős conjectured that for any set A ⊆ N with positive lower asymptotic density, there are infinite sets B, C ⊆ N such that B +C ⊆ A. We verify Erdős' conjecture in the case that A has Banach density exceeding 1 2 . As a consequence, we prove that, for A ⊆ N with positive Banach density (a much weaker assumption than positive lower density), we can find infinite B, C ⊆ N such that B + C is contained in the union of A and a translate of A. Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős' conjecture for subsets of the natural numbers that are pseudorandom.
Abstract. A construction of the real number system based on almost homomorphisms of the integers Z was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction, to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On-saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG. In NBG, it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter).
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