Bromgrev said, Well, your in luck, Class M V and M IV stars are the most common sorts of stars around. A fine example of such a star is Proxima Centauri with 0.00006 times the illumination of the Sun. To get to a distance where the illumination of this star equals that of our sun, you need to reduce the radius from Proxima's center to 1,139,322 km. Proxima is roughly the size of Jupiter or about 150,000 km in diameter. Tidal forces would be fairly strong at this distance, but since a ringworld is not a planet, this is not so important. I'd say a good width for this ringworld would be 20,037 km from rim wall to rim wall, as this represents the distance between Earth's north and south pole if one follows the curves of the Earth. This ringworld's circumference would be 7,158,571 km, or about 179 times Earth's circumference at the equator. The ringworld would need an inner ring not only to provide day and night, but also to shield the ringworld from those stellar flares that often double or triple these red dwarves luminosity for a time. Instead of shadow squares, the inner ring would consist of a grid with a bunch on tiny squares in it that can rotate on hinges 360 degrees, they can shut out all the light from Proxima at a moment's notice, it might also be preferable to have the entire ringworld experience night all at once, that way the ringworld's nights are glared out by reflected light from parts of the ringworld that aren't in shadow.
The rule is basically, the dimmer the star the easier it is to build a ringworld. For one thing you don't need to use as strong material as you would for a G-Class star. Another thing is that you don't need to provide a full 1-g of simulated gravity to retain the atmosphere if you build the walls high enough. A half a g is fine and it reduces the load on the ringworld floor material.