Distribution of unit mass on one fractal self-similar web-type curve

In the article, we study structural, spectral, topological, metric and fractal properties of distribution of complex-valued random variableτ=∑ₙ₌₁∞2ετ3ⁿ≡Δᵍτ₁...τₙ..., where (τₙ) is a~sequence of independent random variables taking the values 0,1,⋯,6 with the probabilities p₀ₙ, p₁ₙ,⋯,p₆ₙ; ε₆=0, ε₀, ε₁,⋯,ε₅ are 6th roots of unity. We prove that the set of values of random variable τ is self-similar six petal snowflake which is a fractal curve G of spider web type with dimension log₃7. Its outline is the Koch snowflake. We establish that τ has either a discrete or a singularly continuous distribution with respect to two-dimensional Lebesgue measure. The criterion of discreteness for the distribution is found and its point spectrum (set of atoms) is described. It is proved that the point spectrum is a countable everywhere dense set of values of the random variable τ, which is the tail set of the seven-symbol representation of the points of the curve G. In the case of identical distribution of the random variables τₙ (namely: pₖₙ=pₖ) we establish that the spectrum of distribution τ is a self-similar fractal and that the essential support of density is the fractal Besicovitch-Eggleston type set. The set is defined by terms digits frequencies and has the fractal dimension α₀(E)=ln p₀p₀⋯ p₆p₆-ln 7 with respect to the Hausdorff-Billingsley α-measure. The measure is a probabilistic generalization of the Hausdorff α-measure. In this case, the random variables τ=Δᵍτ₁⋯τₙ⋯ and τ'=Δᵍτ₁'...τₙ'... defined by different probability vectors (p₀,⋯,p₆) and (p'₀,⋯,p'₆) have mutually orthogonal distributions.