Density of exponentials and Perron-Frobenius operators

In this article, we study the weak-star density of the linear span of the trigonometric functions$\left\{e_{m,n}\right(x,y)=e^{\pi i(mx+ny)},e_{m,n}^{<\beta >}(x,y)=e^{\pi i\beta (m/x+n/y)};m,n\in Z\}$ for a positive real β. We aim to extend the results of Hedenmalm and Montes-Rodríguez (2011) [18] and Canto-Martín, Hedenmalm, and Montes-Rodríguez (2014) [8] in the plane. They have extensively studied the weak-star completeness of the hyperbolic trigonometric system in $L^{\infty }\left(R\right)$. This is the dual formulation of the Heisenberg uniqueness pair for (hyperbola, certain lattice-cross).

As in their work, $\beta =1$ turns out to be the critical value. In particular, one of our main results asserts that the space spanned by the aforesaid trigonometric functions is weak-star dense in $L^{\infty }$ of the set ${\Theta }_{1,\beta }={(-1,1]}^2\cup {(R\setminus (-\beta ,\beta \left]\right)}^2$ if and only if $0<\beta \le 1$, and the corresponding pre-annihilator space has finite dimension whenever $\beta >1$. However, for $\beta >1$, the pre-annihilator space can be made infinite-dimensional by allowing functions with slightly bigger support than ${\Theta }_{1,\beta }$. To be precise, let ${\Theta }_{\beta }^{″}\subseteq R^2\setminus {\Theta }_{1,\beta }$ be such that ${(-\beta ,\beta ]}^2\cap {\Theta }_{\beta }^{″}$ has positive Lebesgue measure. We prove that the weak-star closure of the linear span of $e_{m,n}$ and $e_{m,n}^{<\beta >}$ as $m,n$ varies over $Z$, has infinite codimension in $L^{\infty }({\Theta }_{1,\beta }\cup {\Theta }_{\beta }^{″})$ whenever $\beta >1$. Our proof goes via the analysis of a two-dimensional Gauss-type map and its corresponding Perron-Frobenius operator.