Rotated time-frequency lattices are sets of stable sampling for continuous wavelet systems

We provide an example for the generating matrix $A$ of a two-dimensional lattice $\mathrm{\Gamma }=A{\mathbb{Z}}^2$, such that the following holds: For any sufficiently smooth and localized mother wavelet $\psi$, there is a constant $\beta (A,\psi )>0$, such that $\beta \mathrm{\Gamma }\cap (\mathbb{R}\times {\mathbb{R}}^{+})$ is a set of stable sampling for the wavelet system generated by $\psi$, for all $0<\beta \le \beta (A,\psi )$. The result and choice of the generating matrix are loosely inspired by the studies of low discrepancy sequences and uniform distribution modulo $1$. In particular, we estimate the number of lattice points contained in any axis parallel rectangle of fixed area. This estimate is combined with a recent sampling result for continuous wavelet systems, obtained via the oscillation method of general coorbit theory.