Packing measure formula for regular-polygon self-similar sets

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Let ErE_r be the self-similar set associated with a system of contracting similitudes Ψ={f1,f2,…,fN}\Psi=\{f_1,f_2,\ldots,f_N\} satisfying the strong separation condition. Suppose that, for every i=1,…,Ni=1,\ldots,N, fi(x)=rx+bif_i(x)=rx+b_i, where r∈(0,1N)r\in(0,\frac{1}{N}) and x,bi∈Rnx,b_i\in\mathbb{R}^n. Let s(r)s(r) be the similarity dimension, let μ\mu denote the natural measure, and suppose that the fixed points of the NN similarities are the vertices of an NN-regular polygon. Then

Regular-polygon packing measure formula. The conjectured packing measure satisfies

Ps(r)(Pr)=∣B(xi,1−r)∣s(r)μ(B(xi,1−r))=(21−rr)s(r).P^{s(r)}(P_r)=\frac{|B(x_i,1-r)|^{s(r)}}{\mu(B(x_i,1-r))}=\left(2\frac{1-r}{r}\right)^{s(r)}.

The claim is presented as the conjectural conclusion of the numerical and algorithmic results preceding it. The supplied text does not indicate whether it has been proved or disproved.

References

Primary source

Marta Llorente and Manuel Morán, “Computability of the packing measure of totally disconnected self-similar sets”, arXiv:1404.6920 (2014).

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