On the coexistence of convergence and divergence phenomena for integral averages and an application to the Fourier–Haar series

Let \(C,D\subset \mathbb{N}\) be disjoint sets, and \(\mathcal{C}=\{1/2^{c}\colon c\in C\}, \mathcal{D}=\{1/2^{d}\colon d\in D\}\). We consider the associate bases of dyadic, axis-parallel rectangles \(\mathcal{R}_{\mathcal{C}}\) and \(\mathcal{R}_{\mathcal{D}}\). We give necessary and sufficient conditions on the sets \(\mathcal{C} and \mathcal{D}\) such that there is a positive function \(f\in L^{1}([0,1)^{2})\) so that the integral averages are convergent with respect to \(\mathcal{R}_{\mathcal{C}}\) and divergent for \(\mathcal{R}_{\mathcal{D}}\). We next apply our results to the two-dimensional Fourier--Haar series and characterize convergent and divergent sub-indices. The proof is based on some constructions from the theory of low-discrepancy sequences such as the van der Corput sequence and an associated tiling of the unit square.