We study bounded remainder sets with respect to an irrational rotation of the d-dimensional torus. The subject goes back to Hecke, Ostrowski and Kesten who characterized the intervals with bounded remainder in dimension one.

Sets of bounded discrepancy for multi-dimensional irrational rotation

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Bounded remainder sets

Institut de Mathematiques de Luminy CNRS-UMR 6206, Case 907, 163 av. de Luminy, 13288 Marseille Cedex 9

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Weak mixing and eigenvalues for Arnoux-Rauzy sequences

We consider methods of establishing ergodicity of group extensions, proving that a class of cylinder flows are ergodic, coalescent and non-squashable. A new Koksma-type inequality is also obtained.

Koksma’s Inequality and Group Extensions of Kronecker Transformations

For any irrational cut-and-project setup, we demonstrate a natural infinite family of windows which gives rise to separated nets that are each bounded distance to a lattice. Our proof provides a new construction, using a sufficient condition of Rauzy, of an infinite family of non-trivial bounded remainder sets for any totally irrational toral rotation in any dimension.

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Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices

LetT be the mod 1 circle group, α∈T be irrational and 0<β<1. LetE be the closed subgroup ofR generated by β and 1. DefineX=T×E andT:X→X byT(x, t)=(x+α,t+1[0,β](x)−β). Then we have the theorem:T is ergodic if and only if β is rational or 1, α and β are linearly independent over the rationals.

Ergodicity of cylinder flows arising from irregularities of distribution

LetT be the unit circle, α irrational andF: T → R a step function. A necessary and sufficient condition for the skew of the α-rotation byf (considered as taking values mod 1) to be minimal is given. Also, the boundedness of Σi=1nf(x+iα asn → α is resolved.

Ishai Oren

Papers

Ergodicity of cylinder flows arising from irregularities of distribution

Admissible functions with multiple discontinuities