Dimension dependence and dimension-free ${\mathrm{\ell }}^2$ estimates for variation seminorms of spherical means on the hypercube

We prove that the ${\mathrm{\ell }}^2\to {\mathrm{\ell }}^2$ norm of the $r$-variation seminorm of spherical means on the hypercube admits no dimension-free bounds for any $r\in [1,\mathrm{\infty })$ when the variation is taken over all possible radii. Furthermore, we establish that if the radii are restricted to a fixed parity, dimension-free estimates hold for all $r\in (2,\mathrm{\infty })$.