The 3k−4$3k-4$ theorem modulo a prime: High density for A+B$A+B$

Abstract 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 .