Given an integer qand a polynomial f∈Zq[X]of
degree dwith coefficients in the residue ring Zq=Z/qZ, we obtain new results
concerning the number of solutions to congruences of the form y≡f(x) (modq),
with integer variables lying in some cube Bof 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 solutions to the Vinogradov mean value theorem
when the variables run through rather sparse sets.