The statement:
Let $A\subset \mathbb{R}$ be defined as follows: $x\in A$ if and only if there exists $c>0$ so that $$ |x-j2^{-k}|\geq c2^{-k} $$ holds for all $j\in \mathbb{Z}$ and integers $k\geq 0$.
The negation of the above statement:
$x\in A^{c}$ if and only if for all $c>0$, we have that $$ |x-j2^{-k}|<c2^{-k} $$ holds for all $j\in \mathbb{Z}$ and integers $k\geq 0$.