Existence of a somewhat-smooth number in the interval $[x, x+ \log(x)]$ Smooth numbers in short intervals have been studied deeply in recent years with results of the form that $\psi(x, x^a)-\psi(x-x^b, x^a) \gg x^{b-\epsilon}$ for all $b>1-a-a(1-a)^3$ when $a \in (\frac 1 2, 1)$ (A.Weingartner, Somewhat smooth numbers in short intervals, https://arxiv.org/pdf/2105.13568.pdf). My question is whether there has been much progress on the short intervals to which at least one (somewhat)-smooth number belongs to? Specifically, should we expect at least one $x^{1-\epsilon}$-smooth number in the interval $[x, x+\log(x) ]$ for some $\epsilon > 0$?
 A: It seems that the best we can prove is mentioned in the paper you quoted.
A heuristic argument suggests that for every fixed $0< \epsilon < 1-\exp(-1/e)=0.307799...$ and
all $x>x_0(\epsilon)$, the interval $[x,x+\log x]$ contains at least one $x^{1-\epsilon}$-smooth integer.
On the other hand, for every fixed $\epsilon > 1-\exp(-1/e)$, the interval  $[x,x+\log x]$ contains no $x^{1-\epsilon}$-smooth integer for infinitely many $x \in \mathbb{N}$.
The heuristic argument goes like this: Let $0<\epsilon<1/2$.
Assume each integer in the interval $(n, 2n]$ is $n^{1-\epsilon}$-smooth with probability
$q=\rho(1/(1-\epsilon))=1+\log(1-\epsilon)$,  independent from one another, where $\rho$ is the Dickman function. Let $p=1-q=-\log(1-\epsilon)$. The largest gap
between $n^{1-\epsilon}$-smooth numbers is one more than the longest run of numbers that are not $n^{1-\epsilon}$-smooth.
The longest such run is almost surely $\sim \log(nq)/\log(1/p)$. The threshold value $\epsilon=  1-\exp(-1/e)$ comes from
solving $\log(1/p)=1$ for $\epsilon$. See
The Surprising Predictability of Long Runs
for a discussion of the distribution of the longest run of successes in $n$ independent Bernoulli trials.
