  Denote by $s_0^{(r)}$ the least integer such that if $s \ge s_0^{(r)}$, and
$F$ is a cubic form with real coefficients in $s$ variables that splits into
$r$ parts, then $F$ takes arbitrarily small values at nonzero integral points.
We bound $s_0^{(r)}$ for $r \le 6$.
