The ultimate sparseness occurs when $Ax^*(\lambda)=0$, which is the case when $2\lambda d<\|A\|_{\ell^2\to\ell^1}$. Here $d$ is the Euclidean distance from $b$ to the kernel of $A$. Indeed, the minimizer $x^*$ will be the projection of $b$ onto $\ker A$.
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