Estimates For Integer Points on Algebraic Variety by using Diophantine Approxiwatic

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For transcendental number x, we consider Diophantine approximations by algebraic number beta in terms of height of beta and degree of beta. By using such approximation, called transcendental (or transcendence) measure, we observe integer points on algebraic curves, for example, elliptic curves. We obtain here a refinement for such measure when x comes from coordinates of inverse image by exponential map of rational points of an algebraic group defined over a number field, which is defined as a non trivial extention of a simple abelian variety by the additine group. We get also some finiteness result concerning with integer solutions to certain exponential Diophantine equations.