Abelian groups with layered tiles and the sumset phenomenon

摘要

We prove a generalization of the main theorem in Jin, The sumset phenomenon, 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's man's theorem to illustrate a different approach (which uses the sumset phenomenon) to some combinatorial problems. [References: 8]