The talk will present some recent advances at the crossroads between Number Theory and Fractal Geometry requiring the input of algebraic theories to estimate the measure and/or the factal dimension of sets emerging naturally in Diophantine Approximation. Examples include the proof of metric, uniform and quantitative versions of the Oppenheim conjecture generalised to the case of any homogeneous form and also the determination of the Hausdor dimension of the set of well approximable points lying on polynomially dened manifolds (i.e. on algebraic varieties).[-]