The $3k-4$ Theorem modulo a Prime: High Density for $A+B$

The $3k-4$ Theorem asserts that, if A,\,B⊆ Z are finite, nonempty subsets with |A|≥ |B| and $|A+B|=|A|+|B|+r< |A|+2|B|-3$, then there are arithmetic progressions PA and PB of common difference with X⊆ PX with |PX|≤ |X|+r+1 for all X∈ ,B\. There is much progress extending this result to Z/p Z with p≥ 2 prime. Here we begin by showing that, if A,\,B⊆ G= Z/p Z are nonempty with |A|≥ |B|, A+B≠ G, |A+B|=|A|+|B|+r≤ |A|+1.0527|B|-3, and |A+B|≤ |A|+|B|-9(r+3), then there are arithmetic progressions PA, PB and PC of common difference such that X⊆ PX with |PX|≤ |X|+r+1 for all X∈ ,B,C\, where C=-\,G (A+B). This gives a rare high density version of the $3k-4$ Theorem for general sumsets $A+B$ and is the first instance with tangible (rather than effectively existential) values for the constants for general sumsets $A+B$ with high density. The ideal conjectured density restriction under which a version of the $3k-4$ Theorem modulo p is expected is |A+B|≤ p-(r+3). In part by utilizing the above result as well as several other recent advances, we extend methods of Serra and Z\'emor to give a version valid under this ideal density constraint. We show that, if A,\,B⊆ G= Z/p Z are nonempty with |A|≥ |B|, A+B≠ G, |A+B|=|A|+|B|+r≤ |A|+1.01|B|-3, and |A+B|≤ |A|+|B|-(r+3), then there exist arithmetic progressions PA, PB and PC of common difference such that X⊆ PX with |PX|≤ |X|+r+1 for all X∈ ,B,C\, where C=-\,G (A+B). This notably improves upon the original result of Serra and Z\'emor, who treated the case $A+A$, required p be sufficiently large, and needed the much more restrictive small doubling hypothesis |A+A|≤ |A|+1.0001|A|.