We will give new upper bounds for the number of solutions to the inequalities of the shape
|
F
(
x
,
y
)
|
≤
h
|F(x,y)| \leq h
, where
F
(
x
,
y
)
F(x,y)
is a sparse binary form, with integer coefficients, and
h
h
is a sufficiently small integer in terms of the discriminant of the binary form
F
F
. Our bounds depend on the number of non-vanishing coefficients of
F
(
x
,
y
)
F(x,y)
. When
F
F
is “really sparse”, we establish a sharp upper bound for the number of solutions that is linear in terms of the number of non-vanishing coefficients. This work will provide affirmative answers to a number of conjectures posed by Mueller and Schmidt in [Trans. Amer. Math. Soc. 303 (1987), pp. 241–255], [Acta Math. 160 (1988), pp. 207–247], in special but important cases.