Equidistribution of values of linear forms on a cubic hypersurface

Let C be a cubic form with integer coefficients in n variables, and let h be the h-invariant of C. Let L1,…,Lr be linear forms with real coefficients such that, if α ∈ ℝr ∖{0}, then α⋅L is not a rational form. Assume that h > 16 + 8r. Let τ ∈ ℝr, and let η be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions x ∈ [−P,P]n to the system C(x) = 0, |L(x) −τ| < η. If the coefficients of the linear forms are algebraically independent over the rationals, then we may replace the h-invariant condition with the hypothesis n > 16 + 9r and show that the system has an integer solution. Finally, we show that the values of L at integer zeros of C are equidistributed modulo 1 in ℝr, requiring only that h > 16.