The Sumset Phenomenon

Answering a problem posed by Keisler and Leth, we prove a theorem in nonstandard 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. 1 A theorem in nonstandard analysis Let V be a nonstandard extension of a standard universe V , which contains all standard real numbers. The reader may consult [7] or [3] for basic knowledge of nonstandard analysis. We denote by N the set of all standard non{negative integers, and denote by N the set of all non{negative integers in V . All integers in NrN are called hyper nite integers. For any two sets A and B, and a binary operator we write A B for the set fa b : a 2 A and b 2 Bg. An in nite initial segment U of N is called a cut if U+U U . A cut is an external set (except U = N) because U cannot have a largest element. If U is a cut and c is a real number in V , we write c > U if c is greater than every element in U . Given a hyper nite integer H, we always write H for the set f0; 1; : : : ; H 1g. 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 2 N, U < H=n. Suppose U H is a cut. A set A H is called U{nowhere dense if for any interval [a; b] H of integers such that b a > U , there is a subinterval [c; d] [a; b] r A such that d c > U . U{topology and U{nowhere dense are introduced in [6]. A real number R in V is nite if jRj < n for some standard integer n. Each nite R has a standard part, denoted by st(R), and de ned to be the unique standard real number r such Mathematics Subject Classi cation: Primary 03H05, 03H15. Secondary 11B05, 11B13, 28E05 The research was supported in part by a Ralph E. Powe Junior Faculty Enhancement Award from Oak Ridge Association Universities, a Faculty Research and Development Summer Grant from College of Charleston, and the NSF grant DMS{#0070407.