Immanuel Halupczok

Integrability of oscillatory functions on local fields: transfer principles

For oscillatory functions on local fields coming from motivic exponential functions, we show that integrability over Q n p implies integrability over F p ((t)) n for large p , and vice versa. More generally, the integrability only depends on the isomorphism class of the residue field of the local field, once the characteristic of the residue field is large enough. This principle yields general local integrability results for Harish-Chandra characters in positive characteristic as we show in other work. Transfer principles for related conditions such as boundedness and local integrability are also obtained. The proofs rely on a thorough study of loci of integrability, to which we give a geometric meaning by relating them to zero loci of functions of a specific kind.

Transfer principles for bounds of motivic exponential functions

We study transfer principles for upper bounds of motivic exponential functions and for linear combinations of such functions, directly generalizing the transfer principles from Cluckers and Loeser (Ann Math 171:1011–1065, 2010) and Shin and Templier (Invent Math, 2015, Appendix B). These functions come from rather general oscillatory integrals on local fields, and can be used to describe, e.g., Fourier transforms of orbital integrals. One of our techniques consists in reducing to simpler functions where the oscillation only comes from the residue field.

Definable sets up to definable bijections in Presburger groups: DEFINABLE SETS UP TO DEFINABLE BIJECTIONS IN Z-GROUPS

We entirely classify definable sets up to definable bijections in

A definable Henselian valuation with high quantifier complexity

\mathbb{Z}

LOCAL INTEGRABILITY RESULTS IN HARMONIC ANALYSIS ON REDUCTIVE GROUPS IN LARGE POSITIVE CHARACTERISTIC

-groups, where the language is the one of ordered abelian groups. From this, we deduce, among others, a classification of definable families of bounded definable sets.

Approximations and Lipschitz continuity in p-adic semi-algebraic and subanalytic geometry

We give an example of a parameter-free definable Henselian valuation ring which is neither definable by a parameter-free inline image-formula nor by a parameter-free inline image-formula in the language of rings. This answers a question of Prestel.