Let be a cubic form with integer coefficients in variables, and let be the -invariant of . Let be linear forms with real coefficients such that, if , then is not a rational form. Assume that . Let , and let be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions to the system , . If the coefficients of the linear forms are algebraically independent over the rationals, then we may replace the -invariant condition with the hypothesis and show that the system has an integer solution. Finally, we show that the values of at integer zeros of are equidistributed modulo in , requiring only that .
"Equidistribution of values of linear forms on a cubic hypersurface." Algebra Number Theory 10 (2) 421 - 450, 2016. https://doi.org/10.2140/ant.2016.10.421