In fact quadrupling the number of pixels does not result necessarily in a 4x multiplication in bitrate, much less in general. That is because a movie recorded for example at 4K resolution is not at all the same as 4 different/independent 1080p movies that you would stick together in the same 4K frame. There are correlations/statistical properties that are exploited in the first case by the encoder and that does not exist in the second case. I have seen somewhere (though I can't find the reference) a back of the enveloppe calculation using Fourier analysis that showed that to multiply by 4 the number of pixel you only needed to multiply by 2 the bitrate. So, if HEVC has 50% bitrate savings that would explain the claim that 4K HEVC could have the same bitrate as 1080p AVC. Of course there is much more happening in HEVC than just a Discrete Cosine Transform (motion compensation etc.) so I don't know how this really applies in practice, and I haven't done the tests myself...

The 2x factor reason is pretty intuitive if you understand that video compression is still fundamentally representing superpositioned signals from 2D Fourier Analysis and that multiplying by 2x the number of pixels in each direction is no different for perception than if we doubled the DPI. Double the DPI yields up to 2x the possible frequencies needing to be represented up to the Nyquist frequency that would cover each possible interpolated pixel. This is part of why noise in film or audio makes representation much tougher - noise is typically higher frequency and rather random (although distribution depends upon brown, white, pink, etc. noise).

4x number of points in real space -> 4x points in Fourier space, the Fourier space is still 2D. I don't get your reasoning.

OTH if you increase the DPI you bring in higher frequency components that are not that important for perception, so you can compress them more heavily.

I was confused with a different concept, disregard that part. FFT is by definition reversible for a discrete signal like a quantized image so each pixel must be reversible, so it has to be 4x total space used with no further operation, correct.

Quantization and filtering are the more important parts of the encoder than the FFT / DCTs since the transform is 1:1 reversible. Compression isn't just the mathematical accuracy of a signal when it comes to lossy algorithms as you know. A 720p video upscaled to 1440p should theoretically be exactly the same size for the sake of effective quality but encoders don't care about just the math and apply perceptual filters because simply doubling pixels looks really bad perceptually it turns out.

Another side of it is that video compression, like image compression (and analogously audio compression), works largely by exploiting limitations of human vision. We can't discern high spatial frequencies as precisely as we can discern low frequencies.

It stands to reason that if you shrink those details even smaller, as is the case with 4K, you'll need even less precision for those coefficients.

I don't think a compressed video with 4 times as many pixels will be 4 times as large. Consider you already have one pixel of each 2x2 pixel group, then you can predict very good how the other pixels are looking, so you don't need as much data anymore.

The same holds true for images/graphics. Take a look at the (blk) rust logo at several different sizes: https://github.com/rust-lang/rust-www/tree/master/logos

16x16: 351 Bytes; 32x32: 691 Bytes, factor 1.97 from 16x16; 64x64: 1.41 KB, factor 2.04; 128x128: 3.4 KB, factor 2.41; 256x256: 7.29 KB, factor 2.14; 512x512: 10.9 KB, factor 1.50

So it certainly can both hold true that 4K HEVC encoded video has the same bitrate as 1080p AVC encoded video and that it has 50% savings.

That always sounded a lot like marketing speak.

The 50% improvement that we have got is pretty f*ing impressive.

IIRC, 4k AVC is only twice the bitrate of 1080p AVC (or something like that - I know that the scaling is less than 1:1, though), which would make 4k HEVC have about the same bitrate as 1080p AVC.

nope, the research group has always had the 50% target from h.264 to h.265. Marketers might have had the math wrong ;)