The Marcinkiewicz-Zygmund Property for Riemann Differences with Geometric Nodes

We study when a Riemann difference of order $n$ possesses the Marcinkiewicz-Zygmund (MZ) property: that is, whether the conditions $f(h) = o(h^{n-1})$ and $Df(h) = o(h^n)$ imply $f(h) = o(h^n)$. This implication is known to hold for some classical examples with geometric nodes, such as $\{0, 1, q, \dots, q^{n-1}\}$ and $\{1, q, \dots, q^n\}$, leading to a conjecture that these are the only such Riemann differences with the MZ property. However, this conjecture was disproved by the third-order example with nodes $\{-1, 0, 1, 2\}$, and we provide further counterexamples and a general classification here. We establish a complete analytic criterion for the MZ property by developing a recurrence framework: we analyze when a function $R(h)$ satisfying $D(h) = R(qh) - A R(h)$, together with $D(h) = o(h^n)$ and $R(h) = o(h^{n-1})$, forces $R(h) = o(h^n)$. We prove that this holds if and only if $A$ lies outside a critical modulus annulus determined by $q$ and $n$, covering both $|q| > 1$ and $|q| < 1$ cases. This leads to a complete characterization of all Riemann differences with geometric nodes that possess the MZ property, and provides a flexible analytic framework applicable to broader classes of generalized differences.