On the packing measure of self-similar sets

TL;DR

This paper constructs a specific self-similar set with fractional dimension where the s-dimensional packing measure is zero, addressing a previously open question in fractal geometry.

Contribution

It provides the first example of a linear self-similar set with positive dimension but zero s-dimensional packing measure, answering an open problem.

Findings

Existence of a linear self-similar set with zero packing measure

Dimension of the set is strictly between 0 and 1

Addresses a question posed by Peres and Solomyak

Abstract

Building on a recent result of M. Hochman, we give an example of a linear self-similar set K such that 0 < dim K = s < 1 and P^s(K) = 0. This answers a question of Y. Peres and B. Solomyak.