Explicit Measurable Counterexamples to Finite-Difference Equations

Abstract

We study finite-difference operators of the form \[ D(f,h)=\sum_{k=1}^n a_k f(b_kh), \] where the coefficients may be complex and the nodes are real. Ash, Catoiu, and Csörnyei characterized the finite differences for which \[ D(f,h)=o(h)\quad\Longrightarrow\quad f(h)=o(h) \] holds for every measurable real-valued function. We extend their counterexample theorem to complex-valued functions defined on \(\mathbb C\). Except for the same exceptional four-term configuration, we construct a Lebesgue measurable function \(f\) such that \[ D(f,h)=0\qquad\text{for every }h\in\mathbb C, \] while \(f(h)\not=o(1)\) as \(h\to0\). The counterexamples are supported on countable multiplicative subgroups. The construction is based on zeros of Laurent polynomials and multiplicative characters, and avoids the infinite-dimensional linear-algebra argument used in the earlier proof.