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We show that the discrete Hardy-Littlewood maximal functions associated withthe Euclidean balls in $\mathbb Z^d$ with dyadic radii have bounds independentof the dimension on $\ell^p(\mathbb Z^d)$ for $p\in[2, \infty]$.

On discrete Hardy-Littlewood maximal functions over the balls in $\mathbb Z^d$: dimension-free estimates