INNER APERIODICITIES AND PARTITIONS OF SETS

Let L subset of R(n) be a point lattice of full dimension, P its basic cell, and A subset of or equal to R(n) an arbitrary set. We call D su bset of or equal to P a periodic part of A (mod L) if there are at lea st two u is an element of L such that u = a - x for some a is an eleme nt of A, x is an element of D, and for all such u we have D + u subset of or equal to A. Let B subset of R(n) be a bounded set. The family B := {B-1,B-2,...,} of at most countable many subsets B-i of B is calle d a covering family if U-i greater than or equal to 1 B-i = B. A cover ing family B is called a (weak) partition of B if B-i boolean AND B-j = empty set (all B-i, i greater than or equal to 1, are Lebesgue measu rable and V(B-i boolean AND B-j) = 0) hold for all 1 less than or equa l to i < j, where V is tile Lebesgue measure in R(n). In this article it is shown that there in a close connection between the property of A having no periodic parts (of positive measure) and (weak) partition o f a set B. Some characterizations of both phenomena are proved. The re sults among others improve two basic theorems in the geometry of numbe rs, the theorems of Minkowski-Blichfeldt and Siegel-Bombieri.