### Abelian groups with layered tiles and the sumset phenomenon

Renling Jin, H. Jerome Keisler
2002 Transactions of the American Mathematical Society
We prove a generalization of the main theorem in [4] about the sumset phenomenon in the setting of an abelian group with layered tiles of cell measures. Then we give some applications of the theorem for multi-dimensional cases of the sumset phenomenon. Several examples are given in order to show that the applications obtained are not vacuous and cannot be improved in various directions. We also give a new proof of Shnirel'man's theorem to illustrate a different approach to some combinatorial
more » ... me combinatorial problems which uses the sumset phenomenon. * 2000 Mathematics Subject Classification. Primary 20K99, 60B15, 22A05, 03H05, Secondary 11B05, 26E35, 28E05. † Key words and phrases. abelian group, layered tiles of cell measures, the sumset phenomenon, upper Banach density. In §1 of this paper, we generalize [4, Theorem 1] to the setting of abelian groups with layered tiles of cell measures. The new generalization makes the applications in §2 to multi-dimensional cases easier. These applications show that the description of the sumset phenomenon needs to be adjusted with "measure" replaced by "product measure" and "order-topology" replaced by "product of order-topology". In §3 , we construct several examples to show that the results about the sumset phenomenon in §2 and in [4] cannot be improved in various directions. For example, it was proven in [4] that if A and B are two subsets of the natural numbers with positive upper Banach density, then A+B is piecewise syndetic, which means that there exists a fixed positive integer k such that A + B + {0, 1, . . . , k} is thick. It is natural to ask whether the least such k is related to the upper Banach density of A and the upper Banach density of B. One example constructed in §3 shows that k is not directly related to these upper Banach densities. In §4, a new proof of Shnirel'man's theorem is given, which uses the idea of the sumset phenomenon rather than a finite combinatorial argument. In §5, two questions are raised.