Main techniques to draw a point at non integer coordinates?
Instead of having a binary on/off per pixel you can instead make a heat map. The more points fall within the pixel the brighter the pixel is. Essentially create a 2D histogram which you then renormalize to get the grayscale for the output.

If you want to blur the points then you can move a portion of the brightness to the next pixel based on the remaining fraction after multiplying to the resolution. If it the coordinate is (0.55313, 0.236831) and your resolution is 1000x1000 then 0.87*0.17 goes to 553, 236, 0.87*0.83 goes to 553, 237, 0.13*0.17 goes to 554, 236 and 0.13*0.83 goes to 554, 237. This is distributed linearly but there are other options.