We study infinite iterated functions systems (IIFSs) consisting of bi-Lipschitz mappings instead of conformal contractions, focusing on IFSs that do not satisfy the open set condition. By assuming the logarithmic distortion property and some cardinality growth condition, we obtain a formula for the Hausdorff, box, and packing dimensions of the limit set in terms of certain topological pressure. By assuming, in addition, the weak separation condition, we show that these dimensions are equal to the growth dimension of the limit set.

The universality for the eigenvalue spacing statistics of generalized Wigner matrices was established in our previous work \cite{EYY} under certain conditions on the probability distributions of the matrix elements. A major class of probability measures excluded in \cite{EYY} are the Bernoulli measures. In this paper, we extend the universality result of \cite{EYY} to include the Bernoulli measures so that the only restrictions on the probability distributions of the matrix elements are the subexponential decay and the normalization condition that the variances in each row sum up to one. The new ingredient is a strong local semicircle law which improves the error estimate on the Stieltjes transform of the empirical measure of the eigenvalues from the order $(N \eta)^{-1/2}$ to $(N \eta)^{-1}$. Here $\eta$ is the imaginary part of the spectral parameter in the definition of the Stieltjes transform and $N$ is the size of the matrix.

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