Non-density of integral points in semigroup orbits implies sparsity of multiplicative dependence in higher dimensions, generalizing Northcott-Siegel theorems and following from Vojta's conjecture.
Silverman,Integer points, Diophantine approximation, and iteration of rational maps, Duke Math
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On higher dimensional integrality and multiplicative dependence in semigroup algebraic dynamics
Non-density of integral points in semigroup orbits implies sparsity of multiplicative dependence in higher dimensions, generalizing Northcott-Siegel theorems and following from Vojta's conjecture.