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|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