Almost $k$-th powers in short intervals

Let $k\geq 3$ be a fixed integer and $x$ be a large real number. Let $00.$ In this paper we show that there exists a constant $1/2\leq δ_k(θ)<1$ such that the interval $[x, x + x^{δ_k(θ) +\varepsilon} ]$ contains an integer of the form $n_1n_2 \cdot