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We consider the dynamical system given by a diagonalizable element $a$ of aclosed linear unimodular algebraic subgroup $G$ of the special linear groupover the $p$-adic numbers acting by translation on a finite volume quotient$X$. Assuming that this action is exponentially mixing (e.g.\ if $G$ is simple)we give an effective version (in terms of $K$-finite vectors of the regularrepresentation) of the following statement: If $\mu$ is an $a$-invariantprobability measure with measure-theoretical entropy close to the topologicalentropy of $a$ then $\mu$ is close to the unique $G$-invariant probabilitymeasure of $X$.

Effectivity of Uniqueness of the Maximal Entropy Measure on $p$-adic homogeneous spaces