Khintchine dichotomy for images of self-similar measures and almost random walks

Following the celebrated breakthrough in Bénard-He-Zhang, we prove Khintchine dichotomy for push-forwards of self-similar measures on ${\mathbb{R}}^d$ under uniform $C^{1+\alpha }$ maps. For this, we also study almost random walks on $\mathrm{S}\mathrm{L}(d+1,\mathbb{R})\mathrm{/}\mathrm{S}\mathrm{L}(d+1,\mathbb{Z})$ and prove their effective equidistribution.