Let $mathsf M_{mathsf S}$ denote the strong maximal operator on $mathbb R^n$ and let $w$ be a non-negative, locally integrable function. For $alphain(0,1)$ we define the weighted sharp Tauberian constant $mathsf C_{mathsf S}$ associated with $mathsf M_{mathsf S}$ by $$ mathsf C_{mathsf S} (alpha):= sup_{substack {Esubset mathbb R^n 0<w(E)<+infty}}frac{1}{w(E)}w({xinmathbb R^n:, mathsf M_{mathsf S}(mathbf{1}_E)(x)>alpha}). $$ We show that $lim_{alphato 1^-} mathsf C_{mathsf S} (alpha)=1$ if and only if $win A_infty ^*$, that is if and only if $w$ is a strong Muckenhoupt weight. This is quantified by the estimate $mathsf C_{mathsf S}(alpha)-1lesssim_{n} (1-alpha)^{(cn [w]_{A_infty ^*})^{-1}}$ as $alphato 1^-$, where $c>0$ is a numerical constant; this estimate is sharp in the sense that the exponent $1/(cn[w]_{A_infty ^*})$ can not be improved in terms of $[w]_{A_infty ^*}$. As corollaries, we obtain a sharp reverse Holder inequality for strong Muckenhoupt weights in $mathbb R^n$ as well as a quantitative imbedding of $A_infty^*$ into $A_{p}^*$. We also consider the strong maximal operator on $mathbb R^n$ associated with the weight $w$ and denoted by $mathsf M_{mathsf S} ^w$. In this case the corresponding sharp Tauberian constant $mathsf C_{mathsf S} ^w$ is defined by $$ mathsf C_{mathsf S} ^w alpha) := sup_{substack {Esubset mathbb R^n 0<w(E)<+infty}}frac{1}{w(E)}w({xinmathbb R^n:, mathsf M_{mathsf S} ^w (mathbf{1}_E)(x)>alpha}).$$ We show that there exists some constant $c_{w,n}>0$ depending only on $w$ and the dimension $n$ such that $mathsf C_{mathsf S} ^w (alpha)-1 lesssim_{w,n} (1-alpha)^{c_{w,n}}$ as $alphato 1^-$ whenever $win A_infty ^*$ is a strong Muckenhoupt weight.
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