Recently, I have learned the following surprising result of Hedenhalm and Montes-Rodriguez(2011, Ann. Math. Theorem 3.1): As $n$ ranges over the integers, the functions $e^{\pi inx}$ and $e^{\pi i\beta n/x}$ form a weak-star- spanning system in $L^{\infty}(\mathbb{R})$ if and only if $0<\beta\leq1$. The proof is not long but quite technical when first reading.
I'm curious about what will hapen when condiering the span of their multiplication and division, thus I have the following 'much weaker' question: Does $$ \text{span}\{ e^{\pi itx}, t\in\mathbb{R}\} $$ form a weak-star system in $L^{\infty}(\mathbb{R})$? Furthermore, does $$ \text{span}\{ e^{\pi it(x+\frac{1}{x})}, e^{\pi it(x-\frac{\beta}{x})}, t\in\mathbb{R}\}? $$ form a weak-star system in $L^{\infty}(\mathbb{R})$ for any but fixed $\beta\in\mathbb{R}$?
Two different point here: $e^{\pi it(x+\frac{1}{x})}$ is no longer periodic and $t$ ranges onver $\mathbb{R}$, so I think the situation is quite different, and I guess that the threshold might be whether or not $\beta=\pm 1$.