Abstract :
Let L subset of Rn be a point lattice of full dimension, P its basic cell, and A subset of or equal to Rn an arbitrary set. We call D subset of or equal to P a periodic part of A (mod L) if there are at least two u set membership, variant L such that u = a − x for some a set membership, variant A, x set membership, variant D, and for all such u we have D + u subset of or equal to A. Let B subset of Rn be a bounded set. The family B := {B1, B2, …,} of at most countable many subsets Bi of B is called a covering family if union or logical sumi ≥ 1 Bi = B. A covering family B is called a (weak) partition of B if Bi ∩ Bj = solidus in circle (all Bi, i ≥ 1, are Lebesgue measurable and V(Bi ∩ Bj) = 0) hold for all 1 ≤ i < j, where V is the Lebesgue measure in Rn. In this article it is shown that there is a close connection between the property of A having no periodic parts (of positive measure) and (weak) partition of a set B. Some characterizations of both phenomena are proved. The results among others improve two basic theorems in the geometry of numbers, the theorems of Minkowski-Blichfeldt and Siegel-Bombieri.
