The Rademacher functions are given by $$r_k(t) = \mathrm{sgn}\sin(2^k\pi t), \quad t \in [0,1)$$ for each $k \in \mathbb{Z}_+$. These are basicaly "square-wave functions" of preiod $2,1,1/2, \dots$ and amplitude 1. These functions have many special properties, the main one being that they are independent in a probibalistic sense. I'm however interested in the following.
Let $a_1, a_2, \dots , a_N$ be a real finite sequence and $1 \leq n \leq N$, then $$|\sum_{k=1}^N a_kr_k(t)| \geq |a_n|$$ on a set of measure at least 1/2.
If one can show that $$\sum_{k \neq n} a_kr_k(t)$$ has the same sign as $a_nr_n(t)$ on such a set, then the result would follow by $$|\sum_{k=1}^N a_kr_k(t)| = |\sum_{k \neq n} a_kr_k(t)| + |a_nr_n(t)| \geq |a_n|.$$ If $n=1$ the result follows from the fact that $$\sum_{k=2}^N a_kr_k(t)$$ is non-negative on a set of measure 1/2 and negative on the other half and $r_1(t)=1$. Im not sure how to argue for the other cases however ($n>1$), and I do expect that there might be some simpler way of reasoning. Does anyone see how this could be fixed?
You noted their probabilistic independence.
Given three independent distributions, each of which only takes a finite number of values,
$\mathbb{P}(A+B\geq0\;|\;C\geq0)=\mathbb{P}(A+B\geq0)$.
Exercise!
So for example,
let $\mathbb{P}(A_1+A_3\geq0)=p$
and suppose without loss of generality that $a_2\neq0$.
Then $\mathbb{P}(|A_1+A_2+A_3|\geq|A_2|)$
$\geq \mathbb{P}(A_1+A_3\geq0\;|\;A_2\geq0)\times\mathbb{P}(A_2\geq0)+\mathbb{P}(A_1+A_3<0\;|\;A_2<0)\times\mathbb{P}(A_2<0)$
$=p\times 1/2+(1-p)\times1/2$.