An interesting result. Do you have a link to the proof?
There are k strings of bytelength n.
There are k * 256 strings of bytelength n+1.
k input strings can decompress to at most k output strings
Therefore only k / (k * 256) of length n+1 strings can be compressed by a byte.
The reason for 'almost' is the miniscule number of strings that will compress more than one bye.
An interesting result. Do you have a link to the proof?