### Abstract

We show that problems of existence and characterization of wavelets for non-expanding dilations are intimately connected with the geometry of numbers; more specifically, with a bound on the number of lattice points in balls dilated by the powers of a dilation matrix A∈GL(n,R). This connection is not visible for the well-studied class of expanding dilations since the desired lattice counting estimate holds automatically. We show that the lattice counting estimate holds for all dilations A with |detA|≠1 and for almost every lattice Γ with respect to the invariant probability measure on the set of lattices. As a consequence, we deduce the existence of minimally supported frequency (MSF) wavelets associated with such dilations for almost every choice of a lattice. Likewise, we show that MSF wavelets exist for all lattices and almost every choice of a dilation A with respect to the Haar measure on GL(n,R).

Original language | English |
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Journal | International Mathematics Research Notices |

Volume | 2017 |

Issue number | 23 |

Pages (from-to) | 7264-7291 |

Number of pages | 28 |

ISSN | 1073-7928 |

DOIs | |

Publication status | Published - 2016 |

## Cite this

Bownik, M., & Lemvig, J. (2016). Wavelets for Non-expanding Dilations and the Lattice Counting Estimate.

*International Mathematics Research Notices*,*2017*(23), 7264-7291. https://doi.org/10.1093/imrn/rnw222