Representation of integers by sparse binary forms.

We will give new upper bounds for the number of solutions to the inequalities of the shape \vert F(x,y)\vert \leq h, where F(x,y) is a sparse binary form, with integer coefficients, and h is a sufficiently small integer in terms of the discriminant of the binary form F. Our bounds depend on the number of non-vanishing coefficients of F(x,y). When 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. [ABSTRACT FROM AUTHOR]