Generalized analyticity on the $N$-torus

Abstract:

In dimension $N=2,$ let $v$ be a unit vector of irrational slope. With $T_{2}$ the 2-torus, say $f\in \mathcal {A}_{v}^{+}$ if $f\in L^{1}(\mathbf {T}_{2})$ and $m\cdot v<0$ $\Rightarrow \widehat {f}\left (m\right ) =0$ for $m$ an integral lattice point. Let $G_{v}$ be the one-parameter group generated by $v$ on $T_{2},$ and let $E\subset G_{v}$ be a closed and bounded set. Call $E$ a set of uniqueness for $\mathcal {A}_{v}^{+}\cap C\left (T_{2}\right )$ if $f\in \mathcal {A}_{v}^{+}\cap C\left (T_{2}\right )$ and if$\ f\left (x\right ) =0$ for $x\in E$ implies $f\equiv 0$ on $T_{2}.$ The following result is established$:$ A necessary and sufficient condition that E be a set of uniqueness for $\mathcal {A} _{v}^{+}\cap C\left (T_{2}\right )$ is that E is a set of positive linear measure.