2,325 research outputs found
The -torsion polytope of amenable groups
We introduce the notion of groups of polytope class and show that
torsion-free amenable groups satisfying the Atiyah Conjecture possess this
property. A direct consequence is the homotopy invariance of the -torsion
polytope among -CW-complexes for these groups. As another application we
prove that the -torsion polytope of an amenable group vanishes provided
that it contains a non-abelian elementary amenable normal subgroup.Comment: 21 page
Sets of bounded discrepancy for multi-dimensional irrational rotation
We study bounded remainder sets with respect to an irrational rotation of the
-dimensional torus. The subject goes back to Hecke, Ostrowski and Kesten who
characterized the intervals with bounded remainder in dimension one.
First we extend to several dimensions the Hecke-Ostrowski result by
constructing a class of -dimensional parallelepipeds of bounded remainder.
Then we characterize the Riemann measurable bounded remainder sets in terms of
"equidecomposability" to such a parallelepiped. By constructing invariants with
respect to this equidecomposition, we derive explicit conditions for a polytope
to be a bounded remainder set. In particular this yields a characterization of
the convex bounded remainder polygons in two dimensions. The approach is used
to obtain several other results as well.Comment: To appear in Geometric And Functional Analysi
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