Points on polynomial curves in small boxes modulo an integer

Given an integer $q$ and a polynomial $f\in {\mathbb{Z}}_q[X]$ of degree $d$ with coefficients in the residue ring ${\mathbb{Z}}_q=\mathbb{Z}\mathrm{/}q\mathbb{Z},$ we obtain new results concerning the number of solutions to congruences of the form $$y\equiv f(x)(\mathrm{m}\mathrm{o}\mathrm{d}q),$$ with integer variables lying in some cube $\mathcal{B}$ of side length $H$. Our argument uses ideas of Cilleruelo, Garaev, Ostafe and Shparlinski which reduces the problem to the Vinogradov mean value theorem and a lattice point counting problem. We treat the lattice point problem differently using transference principles from the Geometry of Numbers. We also use a variant of the main conjecture for the Vinogradov mean value theorem of Bourgain, Demeter and Guth and of Wooley which allows one to deal with rather sparse sets.