Almost cubes and fourth powers in short intervals

In this paper, we study how short an interval $[x,x+x^{\theta }]$ contains an integer of the form $n_1{n}_2{n}_3$ and $m_1{m}_2{m}_3{m}_4$ with $n_1\approx n_2\approx n_3$ and $m_1\approx m_2\approx m_3\approx m_4$. The new idea is to adopt a second moment method (usually used for almost all results) to deduce a result for all short intervals.