Population density of stars within $13$ light-years of us This item from Fox News refers to "over 600 stars that came within 13 light-years of the sun". Because of the past-tense "came", they could have meant stars that are not now within 13 light-years of us but were at some time in the past. But that is not made explicit in the article.

It is said that the nearest neighboring star is $4.3$ light-years from us. And $13$ light-years is about $3$ times that. So the volume within that distance of use is about $3^3=27$ times the volume within $4.3$ light-years of us. If there are $600$ stars within that larger volume, then the average number of stars within that smaller volume should be $600/27 \approx 22.$ But we don't have anywhere near $22$ stars within $4.3$ light-years of us.

So are we in a neighborhood that is unusually sparsely populated compared with what is typical in that larger volume?

Or is this about stars formerly near us that aren't now?

Or just confused journalists?
The criteria for sample selection are admittedly not made overly clear in the preprint itself (Wysoczańska et al.). It appears that the authors started with a list based on stars that were previously believed to have come close to the Sun at some point in the past or future. That list was then augmented based on data from Gaia's second data release, to get a total of 820 objects, and the motions of those stars were then integrated backwards and forwards in time from the present, based on a model of the galaxy's gravitational field.

Here's where that "over 600 stars" came from. The authors ended their integration for a given star when either (i) the star went more than 3 kpc from the Sun or (ii) the time reached 50 million years in the past or the future. With those cutoffs in place, the astronomers found that during that timespan (50 million years before or after the present), 647 stars were within 4 parsecs of the Sun at some point in time. That's the figure Fox is referring to.