In this answer, Fabian said that it's enough to have an improved resource in your cultural border in order to gain access to that resource; while I was under the impression you have to actually allocate a citizen to work on the improved resource in order to get it (whether strategic or luxury), so for example you could never gain access to a resource which is farther than 3 tiles from your closest city.
I seem to remember a case where I improved a far resource and didn't get it, but I'm not so sure anymore. Which one is it, then? Do I have to actively work it or is having it within my borders enough? And if it's the latter case, can I get even further resources by using a culture bomb, which seems to be able to expand my borders even beyond 5 tiles from the nearest city?