The article claims that the lens is 2 to 3 times bigger - comparing the SGS3 and SGS4 only. The article lacks.
In point of fact, the aperature is bigger, they went from F/2.6 to F/2.2, S3 to S4.
Here are the real numbers involved -
Lens specs: One vs. SGS4
focal length: 3.82 mm vs. 4.235 mm
aperature: F/2.0 vs. F/2.2
sensor diameter: 1/3.2"(*) vs. 1/3.06"
(* web articles vary as to whether the One sensor is 1/3" or 1/3.2" sensor)
Given that the equation for lens diameter = focal_length/F_number, we get:
One diameter: 1.91 mm (3.82/2.0)
SGS4 diameter: 1.925 mm (4.235/2.2)
The SGS4 lens is .015 mm larger in diameter than the One - that's much less than the width of a human hair (that's 0.100 mm).
But the SGS4 lets in less light (bigger F-stop) onto the same or slightly larger sensor (matters for total area for gathering light).
If you believe in optical relationships, the One is the clear winner, it's letting in more light from the same-sized lens.
But - those are numbers.
The lens matters. The sensor's rated electrical noise matters.
More discussion on the megapixel myth can be found in the HTC One forum - Camera (and motion video) talk - the Ultrapixel and more (tips/discussion)
Discussion of HTC weaknesses can be found at iPhone 4S to HTC One (or SGS4)
PS - the SGS3 has a focal length of 3.7 mm, F/2.6, lens diameter is 1.423 mm
SGS4 vs. SGS3, in percent difference is - 35% larger diameter, or, 83% more area.
No way the SGS4 lens is 2 to 3 times bigger than the SGS3, the article is just pants.