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We show that under natural technical conditions, the sum of a $C^2$dynamically defined Cantor set with a compact set in most cases (for almost allparameters) has positive Lebesgue measure, provided that the sum of theHausdorff dimensions of these sets exceeds one. As an application, we show thatfor many parameters, the Square Fibonacci Hamiltonian has spectrum of positiveLebesgue measure, while at the same time the density of states measure ispurely singular.

Sums of regular Cantor sets of large dimension and the Square Fibonacci Hamiltonian