Distribution of solutions to systems of congruences in balls

Let G₁, , Gₙ FₚX₁, , Xₘ be n polynomials in m variables over the finite field Fₚ of p elements. For any sufficiently large prime p and non-trivial bounds for the Weyl sums associated to the non-trivial linear combinations of G= (G₁, , Gₙ), we study various properties regarding the distribution of the vectors by fractional parts equation* (\ G₁ ({x) p\}, , \ Gₙ ({x) p\}) Tⁿ, 10pt x Fₚᵐ. equation* We prove refinements of equidistribution, such as bounds for the ball discrepancy and variance.