Positive proportion of short intervals containing a prescribed number of primes

Abstract: We will prove that for every $m\geq 0$ there exists an $\varepsilon=\varepsilon(m)>0$ such that if $0<\lambda<\varepsilon$ and $x$ is sufficiently large in terms of $m$ and $\lambda$, then $$|\lbrace n\leq x: |[n,n+\lambda\log n]\cap \mathbb{P}|=m\rbrace|\gg_{m,\lambda} x.$$ The value of $\varepsilon(m)$ and the implicit constant on $\lambda$ and $m$ may be made explicit. This is an improvement of an author's previous result. Moreover, we will show that a careful investigation of the proof, apart from some slight changes, can lead to analogous estimates when considering the parameters $m$ and $\lambda$ to vary as functions of $x$ or restricting the primes to belong to specific subsets.