Dimension-free maximal inequalities for lattice power balls and exact spheres

We develop an approximation theorem for dimension-free maximal inequalities over lattice sublevel sets. Theorem X applies to general defining functions \(Φ_d: Z^d→[0,∞)\): maximal bounds for suitable approximating operators, together with summable remainders of arbitrary order, give bounds on every \(^p\), \(1 3/2\). For functions of the form \(Φ_d(y)=∑_jφ(y_j)\), Theorem A gives sufficient conditions for X and yields full-radius ball bounds for every fixed real \(1≤ q<∞\), with \(φ(k)=|k|^q\). Theorem B transfers such ball estimates to exact lattice spheres using a spherical maximal estimate and representation-count bounds in one fixed dimension. It gives dimension-free sphere bounds for every finite integer \(q≥2\) and every \(1 1\), the sphere maximal operator is unbounded in every fixed dimension on \(^p\), \(1≤ p<∞\). Thus balls are bounded for all fixed \(1≤ q≤∞\) and \(1<p<∞\), whereas the corresponding high-dimensional spherical bounds hold exactly for finite integers \(q≥2\). All suprema include every radius with a nonempty lattice ball or sphere. At \(p=∞\), both maximal operators are contractions for every \(q\).