Substituting the typical compact sets into a power series

TL;DR

This paper explores the effect of substituting compact sets into power series, revealing conditions under which the resulting set has Hausdorff dimension 0 or 1, extending previous polynomial results.

Contribution

It generalizes prior work on polynomial substitution to broader classes of power series, identifying when the resulting sets have Hausdorff dimension 0 or 1.

Findings

Substituting typical compact sets into certain power series yields sets of Hausdorff dimension 0.

Under different conditions, the substitution results in sets with Hausdorff dimension 1.

The paper extends known results from polynomials to more general power series.

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.