Let $X \subset \mathbf{ P}_{ \mathbf{ Q}}^{ n-1}$ be a cubic hypersurface cut out by the vanishing of a non-degenerate rational cubic form in $n$ variables. Let $N(X,B)$ denote the number of rational points on $X$ of height at most $B$. In this article we obtain lower bounds for $N(X,B)$ for cubic hypersufaces, provided only that $n$ is large enough. In particular, we show that $N(X,B) \gg B^{ n-9}$ if $n \geq 39$, thereby proving a conjecture of T. D. Wooley for non-conical cubic hypersurfaces with large enough dimension.
Categories: math.NT