Let 𝕋 = {a n } n ∪{0} be a time scale with zero Minkowski (or box) dimension, where {a n } n is a monotonically decreasing sequence converging to zero, and a 1 = 1. In this paper, we find an upper bound for the eigenvalue counting function of the linear problem − u ΔΔ = λu σ, with Dirichlet boundary conditions. We obtain that the nth-eigenvalue is bounded below by . We show that the bound is optimal for the q-difference equations arising in quantum calculus.