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2021 Substituting the typical compact sets into a power series
Donát Nagy
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Real Anal. Exchange 46(1): 149-162 (2021). DOI: 10.14321/realanalexch.46.1.0149

Abstract

The Minkowski sum and Minkowski product can be considered as the addition and multiplication of subsets of $\mathbb R$. If we consider a compact subset $K \subseteq [0,1]$ and a power series $f$ which is absolutely convergent on $[0, 1]$, then we may use these operations and the natural topology of the space of compact sets to substitute the compact set $K$ into the power series $f$. Changhao Chen studied this kind of substitution in the special case of polynomials and showed that if we substitute the typical compact set $K \subseteq [0,1]$ into a polynomial, we get a set of Hausdorff dimension 0. We generalize this result and show that the situation is the same for power series where the coefficients converge to zero quickly. On the other hand we also show a large class of power series where the result of the substitution has Hausdorff dimension one.

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Donát Nagy. "Substituting the typical compact sets into a power series." Real Anal. Exchange 46 (1) 149 - 162, 2021. https://doi.org/10.14321/realanalexch.46.1.0149

Information

Published: 2021
First available in Project Euclid: 14 October 2021

Digital Object Identifier: 10.14321/realanalexch.46.1.0149

Subjects:
Primary: 28A78
Secondary: 28A80 , 37F40

Keywords: Baire category , Hausdorff dimension , Hausdorff metric , Minkowski product , Minkowski sum

Rights: Copyright © 2021 Michigan State University Press

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Vol.46 • No. 1 • 2021
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