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

The $3k-4$ Theorem asserts that, if $A,\,B\subseteq \mathbb Z$ are finite, nonempty subsets with $|A|\geq |B|$ and $|A+B|=|A|+|B|+r< |A|+2|B|-3$, then there are arithmetic progressions $P_A$ and $P_B$ of common difference with $X\subseteq P_X$ with $|P_X|\leq |X|+r+1$ for all $X\in \{A,B\}$. There is much progress extending this result to $\mathbb Z/p\mathbb Z$ with $p\geq 2$ prime. Here we begin by showing that, if $A,\,B\subseteq G=\mathbb Z/p\mathbb Z$ are nonempty with $|A|\geq |B|$, $A+B\neq G$, $|A+B|=|A|+|B|+r\leq |A|+1.0527|B|-3$, and $|A+B|\leq |A|+|B|-9(r+3)$, then there are arithmetic progressions $P_A$, $P_B$ and $P_C$ of common difference such that $X\subseteq P_X$ with $|P_X|\leq |X|+r+1$ for all $X\in \{A,B,C\}$, where $C=-\,G\setminus (A+B)$. 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 $|A+B|\leq p-(r+3)$. 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 $A,\,B\subseteq G=\mathbb Z/p\mathbb Z$ are nonempty with $|A|\geq |B|$, $A+B\neq G$, $|A+B|=|A|+|B|+r\leq |A|+1.01|B|-3$, and $|A+B|\leq |A|+|B|-(r+3)$, then there exist arithmetic progressions $P_A$, $P_B$ and $P_C$ of common difference such that $X\subseteq P_X$ with $|P_X|\leq |X|+r+1$ for all $X\in \{A,B,C\}$, where $C=-\,G\setminus (A+B)$. 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 $|A+A|\leq |A|+1.0001|A|$.