Shortest Distance in Modular Cubic Polynomials

In this paper, we study how small a box contains at least two points from a modular cubic polynomial $y\equiv a{x}^3+b{x}^2+cx+d(\mathrm{m}\mathrm{o}\mathrm{d}p)$ with $(a, p) = 1$\mathrm{.}Weprovethatsomesquareofsidelength$p^{1/6 + \epsilon}$ contains two such points.