Abstract Let $X$ be a measure space with measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem understanding structure measurable tilings $F \odot A = X$ by tile $A \subset translated finite set G$ shifts, thus translates $f A$, \in F$ partition up to null sets. Adapting arguments from previous literature, we establish “dilation lemma” that asserts...

The articles [11], [6], [13], [14], [4], [5], [2], [10], [8], [3], [7], [12], [9], and [1] provide the notation and terminology for this paper. For simplicity, we follow the rules: x is a set, i, j, n, m are natural numbers, D is a non empty set, K is a non empty double loop structure, s is a finite sequence, a, a1, a2, b1, b2, d are elements of D, p, p1, p2 are finite sequences of elements of ...

Let G be a nite group and k an algebraically closed eld of characteristic p > 0. If B is a block of the group algebra kG with defect group D, the Brauer correspondent of B is a block b of kN G (D). When D is abelian, the blocks B and b, although they are rarely isomorphic or even Morita equivalent, seem to be very closely related. For example, Alperin's Weight Conjecture predicts that they shou...

We give a set of axioms to establish a perpendicularity relation in anAbelian group and then study the existence of perpendicularities in (Z n , +) and (Q + , ⋅) and in certain other groups. Our approach provides a justification for the use of the symbol⊥ denoting relative primeness in number theory and extends the domain of this convention to some degree. Related to that, we also consider para...