According to this question every interval $[x, x + x^{0.45}]$ contains a product of two primes, and this has been improved further slightly. Are there better results available for $k$-almost primes? What is the smallest constant $c_k$ for which we know that $[x, x + x^{c_k}]$ always contains a $k$-almost prime?

I had hoped that $c_k \rightarrow 0$ but this can't be true if we don't know it for squarefree numbers (unconditionally) yet.

Edit: $c_k \rightarrow 0$ is possible using the linear sieve, which shows something like $[x, x + x^\theta]$ always contains a number not divisible by any primes less than $x^{\theta/2}$ (I might be getting the exponent wrong?). However these numbers are actually a thin subset of all $k$-almost primes, so there seems like there should be some hope of improving this bound.

Also, minor question: this problem is referred to in the literature as both "large gaps between almost primes" and "almost primes in short intervals." Is there any difference?



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