Barak Weiss

Spatial Aliasing in Spherical Microphone Arrays

Performance of microphone arrays at the high-frequency range is typically limited by aliasing, which is a result of the spatial sampling process. This paper presents analysis of aliasing for spherical microphone arrays, which have been recently studied for a range of applications. The paper presents theoretical analysis of spatial aliasing for various sphere sampling configurations, showing how high-order spherical harmonic coefficients are aliased into the lower orders. Spatial antialiasing filters on the sphere are then introduced, and the performance of spatially constrained filters is compared to that of the ideal antialiasing filter. A simulation example shows how the effect of aliasing on the beam pattern can be reduced by the use of the antialiasing filters

On the construction of minimal skew products

It is shown that under fairly general conditions on a compact metric spaceY there are minimal homeomorphisms onZ×Y of the form(z,y)→(σz, hz(y)) where (Z, σ) is a arbitrary metric minimal flow andz→hz is a continuous map fromZ to the space of homeomorphisms ofY. Similar results are obtained for strict ergodicity, topolotical weak mixing and some relativized concepts.

Badly approximable vectors on fractals

For a large class of closed subsetsC of ℝn, we show that the intersection ofC with the set of badly approximable vectors has the same Hausdorff dimension asC. The sets are described in terms of measures they support. Examples include (but are not limited to) self-similar sets such as Cantor’s ternary sets or attractors for irreducible systems of similarities satisfying Hutchinson’s open set condition.

On sets invariant under the action of the diagonal group

We consider the action of the (n-1) -dimensional group of diagonal matrices in SL (n,\mathbb{R}) on SL (n,\mathbb{R})/\Gamma , where \Gamma is a lattice and n\ge 3 . Far-reaching conjectures of Furstenberg, Katok–Spatzier and Margulis suggest that there are very few closed invariant sets for this action. We examine the closed invariant sets containing compact orbits. For example, for \Gamma={\rm SL}(n,\mathbb{Z}) we describe all possible orbit-closures containing a compact orbit. In marked contrast to the case n=2 , such orbit-closures are necessarily homogeneous submanifolds in the sense of Ratner.

Closed orbits for actions of maximal tori on homogeneous spaces

Let G be a real algebraic group defined over Q, let 0 be an arithmetic subgroup, and let T be any torus containing a maximal R-split torus. We prove that the closed orbits for the action of T on G/0 admit a simple algebraic description. In particular, we show that if G is reductive, an orbit T x0 is closed if and only if x−1T x is a product of a compact torus and a torus defined over Q, and it is divergent if and only if the maximal R-split subtorus of x−1T x is defined over Q and Q-split. Our analysis also yields the following: • there is a compact K ⊂ G/0 which intersects every T -orbit; • if rankQ G < rankR G, there are no divergent orbits for T .

THE SET OF BADLY APPROXIMABLE VECTORS IS STRONGLY C 1 INCOMPRESSIBLE

We prove that the countable intersection of C 1 -diffeomorphic images of cer- tain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in R d , improving earlier results of Schmidt and Dani. To prove this, inspired by ideas of McMullen, we define a new variant of Schmidts (�,�)-game and show that our sets are hyperplane absolute winning (HAW), which in particular implies winning in the original game. The HAW property passes automati- cally to games played on certain fractals, thus our sets intersect a large class of fractals in a set of positive dimension. This extends earlier results of Fishman to a more general set-up, with simpler proofs. ∞ \ i=1 f −1 i (S)

Almost no points on a Cantor set are very well approximable

We prove that almost no numbers in Cantors middle–thirds set are very well approximable by rationals. More generally, we discuss Diophantine properties of almost every point, where ‘almost every’ is understood relative to any measure on the real line satisfying a decay condition introduced in the work of Veech.

Bounded geodesics in moduli space

In the moduli space of quadratic differentials over complex structures on a surface, we construct a set of full Hausdorff dimension of points with bounded Teichmuller geodesic trajectories. The main tool is quantitative nondivergence of Teichmuller horocycles, due to Minsky and Weiss. This has an application to billiards in rational polygons.

Ergodicity for infinite periodic translation surfaces

For a Z-cover of a translation surface, which is a lattice surface, and which admits infinite strips, we prove that almost every direction for the straightline flow is ergodic.

Equivalence relations on separated nets arising from linear toral flows

In 1998, Burago-Kleiner and McMullen independently proved the existence of separated nets in