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.