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.