  We consider decoupling for a fractal subset of the parabola. We reduce
studying $l^{2}L^{p}$ decoupling for a fractal subset on the parabola $\{(t,
t^2) : 0 \leq t \leq 1\}$ to studying $l^{2}L^{p/3}$ decoupling for the
projection of this subset to the interval $[0, 1]$. This generalizes the
decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the
sparsity and fractal like structure, this allows us to improve upon
Bourgain-Demeter's decoupling theorem for the parabola. In the case when $p/3$
is an even integer we derive theoretical and computational tools to explicitly
compute the associated decoupling constant for this projection to $[0, 1]$. Our
ideas are inspired by the recent work on ellipsephic sets by Biggs using nested
efficient congruencing.
