We show that a multiplicative form of Dirichlet’s theorem on simultaneous Diophantine approximation as formulated by Minkowski, cannot be improved for almost all points on any analytic curve on R which is not contained in a proper affine subspace. Such an investigation was initiated by Davenport and Schmidt in the late sixties. The Diophantine problem is then settled via showing that certain se...

We define the notion of factorizable quasi-Hopf algebra by using a categorical point of view. We show that the Drinfeld double D(H) of any finite dimensional quasi-Hopf algebra H is factorizable, and we characterize D(H) when H itself is factorizable. Finally, we prove that any finite dimensional factorizable quasi-Hopf algebra is unimodular. In particular, we obtain that the Drinfeld double D(...

We show that Ln is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart h∗-polynomial is given by the (n − 1)st Eulerian polynomial, and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for Ln, we conclude with observations and a conjecture regarding the structure of unimodular triangulati...

We consider converses to the density theorem for square-integrable, irreducible, projective, unitary group representations restricted lattices using dimension theory of Hilbert modules over twisted von Neumann algebras. show that restriction such a σ-projective representation π unimodular, second-countable G lattice Γ extends module algebra (Γ,σ). then compute center-valued this module. For abe...

Unimodular relativity is a theory of gravity and space–time with a fixed absolute space–time volume element, the modulus, which we suppose is proportional to the number of microscopic modules in that volume element. In general relativity an arbitrary fixed measure can be imposed as a gauge condition, while in unimodular relativity it is determined by the events in the volume. Since this seems t...

An important conjecture in percolation theory is that almost surely no infinite cluster exists in critical percolation on any transitive graph for which the critical probability is less than 1. Earlier work has established this for the amenable cases Z and Z for large d, as well as for all non-amenable graphs with unimodular automorphism groups. We show that the conjecture holds for several cla...

We conjecture two generalisations of Elkies’ theorem on unimodular quadratic forms to non-unimodular forms. We give some evidence for these conjectures including a result for determinant 3. These conjectures, when combined with results of Frøyshov and of Ozsváth and Szabó, would give a simple test of whether a rational homology 3-sphere may bound a negative-definite four-manifold. We verify som...