"In this talk, we shall discuss the weak-star density of the linear span of the trigonometric system $\left\{ e_{m,n}(x,y)=e^{\pi i(mx+ny)},~e_{m,n}^{<\beta>}(x,y)=e^{\pi i \beta(m/x+n/y)};~m,n \in \mathbb{Z} \right\}$ in a ``part'' of $\mathbb{R}^2,$ for $\beta>0.$ This has a natural connection with the Heisenberg uniqueness pair. A two-dimensional Gauss-type map and its corresponding Perron-Frobenius operator is in the centre part of our analysis. (Joint work with Dr. Debkumar Giri). "