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

Let \(C,D\subset \mathbb{N}\) $C,D\subset \mathbb{N}$ be disjoint sets, and \(\mathcal{C}=\{1/2^{c}\colon c\in C\}, \mathcal{D}=\{1/2^{d}\colon d\in D\}\) $\mathcal{C}=\{1/2^c:c\in C\},\mathcal{D}=\{1/2^d:d\in D\}$ . We consider the associate bases of dyadic, axis-parallel rectangles \(\mathcal{R}_{\mathcal{C}}\) ${\mathcal{R}}_{\mathcal{C}}$ and \(\mathcal{R}_{\mathcal{D}}\) ${\mathcal{R}}_{\mathcal{D}}$ . We give necessary and sufficient conditions on the sets \(\mathcal{C} and \mathcal{D}\) $\mathcal{C}and\mathcal{D}$ such that there is a positive function \(f\in L^{1}([0,1)^{2})\) $f\in L^1({[0,1)}^2)$ so that the integral averages are convergent with respect to \(\mathcal{R}_{\mathcal{C}}\) ${\mathcal{R}}_{\mathcal{C}}$ and divergent for \(\mathcal{R}_{\mathcal{D}}\) ${\mathcal{R}}_{\mathcal{D}}$ . 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.