The theorem modulo a prime: High density for

The $$ Theorem for $$ asserts that, if $$ are finite, nonempty subsets with $$ and $$, then there exist arithmetic progressions $$ and $$ of common difference such that $$ and $$ for all $$. These are instances of Freiman's theorem with precise bounds. There is much partial progress extending this result to nonempty subsets $$ with $$ prime, $$ and $$. The ideal conjectured density restriction under which such a version of the $$ Theorem modulo $$ is expected is $$. Under this ideal density constraint, we show that there are arithmetic progressions $$, $$, and $$ of common difference with $$ and $$ for all $$, where $$, provided $$. This generalizes a result of Serra and Zémor [33] by extending their work from the special case $$ to that of general sumsets $$, removes all unnecessary sufficiently large $$ restrictions, and improves (even in the case $$) their constant 100-fold, from 0.0001 to 0.01. As part of the proof, we additionally obtain a yet better 1000-fold improvement of their constants at the cost of a near optimal density restriction of the form $$ (Theorem 3.5 and Corollary 3.7). These give rare high-density versions of the $$ Theorem for general sumsets $$ modulo $$ and are the first instances with tangible (rather than effectively existential) values for the constants for general sumsets $$ with high density, or indeed for any density without added constraints on the relative sizes of $$ and $$.