  We prove that the discrete spherical maximal functions (in the spirit of
Magyar, Stein and Wainger) corresponding to the Euclidean spheres in $\mathbb
Z^d$ with dyadic radii have $\ell^p(\mathbb Z^d)$ bounds for all $p\in[2,
\infty]$ independent of the dimensions $d\ge 5$. An important part of our
argument is the asymptotic formula in the Waring problem for the squares with a
dimension-free multiplicative error term. By considering new approximating
multipliers we will show how to absorb an exponential in dimension (like $C^d$
for some $C>1$) growth in norms arising from the sampling principle of Magyar,
Stein and Wainger, and ultimately deduce dimension-free estimates for the
discrete spherical maximal functions.
