Multifractal Structure of Irregular Sets via Weighted Random Sequences

We study the multifractal structure of irregular sets arising from Fibonacci-weighted sums of sequences of random variables. Focusing on Cantor-type subsets Kε of the unit interval, we construct sequences of free and forced blocks, where the free blocks allow full binary branching and the forced blocks fix the digits, controlling the weighted averages. We prove that these sets can attain full Hausdorff and packing dimension while their Hausdorff measure can vanish. We prove that the packing measure of Kϵ depends sensitively on the growth of the forced blocks. Our construction illustrates the mechanism by which Fibonacci-type weights induce irregularity, providing a probabilistic counterpart to classical multifractal phenomena in dynamical systems.