{ "abstract": "Let $d\\ge5$, and let $A_λ$ denote the normalized discrete spherical average on $\\mathbb{Z}^d$ with squared radius $λ$. For a strictly increasing sequence $(μ_k)$ of positive integers, set $λ_k=μ_k!$ and consider the lacunary maximal operator $A_\\star f:=\\sup_k|A_{λ_k}f|$. For each fixed $γ>0$, we prove that the estimate \\[ \\#\\{x:A_\\star f(x)>α\\} \\le K\\sum_x\\frac{|f(x)|}α \\left(1+\\log^+\\frac{|f(x)|}α\\right)^γ\\] holds with some finite $K$ for all $α>0$ and finitely supported $f$ if and only if \\[ \\sup_{N\\ge2}\\frac{\\#\\{k:μ_k\\le N\\}}{(1+\\log N)^γ}<\\infty. \\] The summand is interpreted as zero when $f(x)=0$. The case $γ=1$ answers the $\\ell\\log\\ell$ endpoint question of Kesler, Lacey and Mena in the formulation stated above. For $μ_k=2^{2^k}$, these estimates hold for every fixed $γ>0$, but $A_\\star$ is not of weak type $(1,1)$. For factorial blocks formed from arbitrary subsets of dyadic grids at widely separated scales, we characterize restricted weak type $(p,p)$, $1p_*$, and unbounded on $\\ell^p$ for $1\\le p