Schmidt's game on Hausdorff metric and function spaces: generic dimension of sets and images

Ábel Farkas; Jonathan Fraser; Erez Nesharim; David Simmons

doi:10.1112/mtk.12068

Schmidt's game on Hausdorff metric and function spaces: generic dimension of sets and images

Ábel Farkas, Jonathan Fraser, Erez Nesharim, David Simmons

Pure Mathematics

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Abstract

We consider Schmidt's game on the space of compact subsets of a given metric space equipped with the Hausdorff metric, and the space of continuous functions equipped with the supremum norm. We are interested in determining the generic behavior of objects in a metric space, mostly in the context of fractal dimensions, and the notion of “generic” we adopt is that of being winning for Schmidt's game. We find properties whose corresponding sets are winning for Schmidt's game that are starkly different from previously established, and well‐known, properties which are generic in other contexts, such as being residual or of full measure.

Original language	English
Pages (from-to)	196-213
Journal	Mathematika
Volume	67
Issue number	1
DOIs	https://doi.org/10.1112/mtk.12068
Publication status	Published - 29 Dec 2020

Keywords

Schmidt's game
Dimension

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Farkas_2020_Mathematika_Schmidt'sgame_AAM
Copyright © 2020 The Authors. The publishing rights for this article are licensed to University College London under an exclusive licence. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1112/mtk.12068
Accepted author manuscript, 288 KB

https://arxiv.org/abs/1907.07394

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. J. (CoI)
EPSRC
1/02/18 → 11/06/21
Project: Standard
Fractal Geometry and Dimension: Fractal Geometry and dimension theory
Fraser, J. (PI)
The Leverhulme Trust
1/09/16 → 30/06/18
Project: Fellowship

Cite this

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author = "{\'A}bel Farkas and Jonathan Fraser and Erez Nesharim and David Simmons",

note = "AF was financially supported by an ERC Consolidator Grant (772466) and by The MTA Momentum Project (LP2016‐5). JMF was financially supported by a Leverhulme Trust Research Fellowship (RF‐2016‐500) and an EPSRC Standard Grant (EP/R015104/1). EN and DS were supported by an EPSRC Programme Grant (EP/J018260/1). ",

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N1 - AF was financially supported by an ERC Consolidator Grant (772466) and by The MTA Momentum Project (LP2016‐5). JMF was financially supported by a Leverhulme Trust Research Fellowship (RF‐2016‐500) and an EPSRC Standard Grant (EP/R015104/1). EN and DS were supported by an EPSRC Programme Grant (EP/J018260/1).

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Schmidt's game on Hausdorff metric and function spaces: generic dimension of sets and images

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Keywords

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Projects

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Fractal Geometry and Dimension: Fractal Geometry and dimension theory

Cite this