Suppose that $\varepsilon_i$ are independent Rademacher random variables (that is, $ \mathbb{P}(\varepsilon_i=-1) = \mathbb{P}(\varepsilon_i=1) =1/2 $. Fix an $a\in\mathbb{R}^n$ and define the random variable $X=\sum_{i=1}^n a_i\varepsilon_i$.

I am interested in lower and upper bounds on $\mathbb{E}e^{-|X|/2}$ of the form $$ L(a) \le \mathbb{E}\exp\left(-\frac12|X|\right) \le U(a) $$ that satisfy the bounded ratio property: $$ \sup_{a\in\mathbb{R}^n} \frac{U(a)}{L(a)} <\infty. $$

Note that the supremum is over $\mathbb{R}^n$ for a fixed $n$ and will almost certainly depend on $n$.

Any ideas much welcome.