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Given an integer $q$ and a polynomial $f\in \mathbb Z_{q}[X]$ of degree $d$with coefficients in the residue ring $\mathbb Z_q=\mathbb Z/q\mathbb Z,$ weobtain new results concerning the number of solutions to congruences of theform $$y\equiv f(x) \pmod{q},$$ 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 valuetheorem and a lattice point counting problem. We treat the lattice pointproblem differently using transference principles from the Geometry of Numbers.We also use a variant of the main conjecture for the Vinogradov mean valuetheorem of Bourgain, Demeter and Guth and of Wooley which allows one to dealwith rather sparse sets.

Points on polynomial curves in small boxes modulo an integer