Let
A
\mathcal {A}
be an ellipsephic set which satisfies digital restrictions in a given base. Using the method developed by Hughes and Wooley, we bound the number of integer solutions to the system of equations
a
m
p
;
∑
i
=
1
2
(
x
i
3
−
y
i
3
)
=
∑
i
=
3
5
(
x
i
3
−
y
i
3
)
a
m
p
;
∑
i
=
1
2
(
x
i
−
y
i
)
=
∑
i
=
3
5
(
x
i
−
y
i
)
,
\begin{equation*} \begin {split} & \sum _{i=1}^2 \left (x_i^3-y_i^3 \right )=\sum _{i=3}^5 \left (x_i^3-y_i^3 \right )\\ & \sum _{i=1}^2 (x_i-y_i)=\sum _{i=3}^5 (x_i-y_i), \end{split} \end{equation*}
with
x
,
y
∈
A
5
\mathbf {x, y} \in \mathcal {A}^5
. The fact that ellipsephic sets with small digit sumsets have fewer solutions of linear equations allows us to improve the general bounds obtained by Hughes and Wooley and also the corresponding efficient congruencing estimates. We also generalize our result from the curve
(
x
,
x
3
)
(x, x^3)
to
(
x
,
ϕ
(
x
)
)
(x, \phi (x))
, where
ϕ
\phi
is a polynomial with integer coefficients and
deg
(
ϕ
)
≥
3
\deg (\phi ) \ge 3
.