On the critical pair theory in Z/pZ

Abstract

Let A, B be subsets of Z/pZ such thatWe prove that, if |A| ≥ 4, |B| ≥ 5, |A + B| ≤ p − 5 and p ≥ 53 then A and B are included in arithmetic progressions with the same difference and of size |A| + 2 and |B| + 2 respectively. This extends the well-known theorem of Vosper and a recent result of Rødseth and one of the present authors.The Cauchy-Davenport Theorem [2,4] states that if A and B are subsets of Z/pZ then |A + B| ≥ min(p, |A| + |B| − 1). Vosper's Theorem [21] solves the related critical pair problem and states that, if |A|, |B| ≥ 2, then |A + B| ≥ min(p − 1, |A| + |B|)

A Step Beyond Kemperman's Structure Theorem

“…The extent to which the result holds in Z/ pZ without unnecessary assumptions on the cardinalities is still quite open. Little is known beyond the case |A + B| = |A| + |B|, for which Hamidoune and Rødseth established d ⊆ (A, AP d ), d ⊆ (B, AP d ) ≤ 1, with only the assumption |A + B| ≤ p − 4 and the removal of ǫ from (1) (recall |A| ≥ |B|) [22]; and the case |A + B| = |A| + |B| + 1, for which d ⊆ (A, AP d ), d ⊆ (B, AP d ) ≤ 2 was shown by Hamidoune et al under similar assumptions with p > 51 [23]. Concerning slightly more general abelian groups, for A ⊆ Z/m Z with |A + A| < 2.04|A|, Deshouillers and Freiman obtained a rough description of A involving computed but very large constants [3].…”

“…imply |H 1 | = 1 also. Thus(23) implies |A| ≤ 2, which contradicts the hypothesis.LEMMA 5.3. Let A and B be finite, non-empty subsets of an abelian group G with A + B aperiodic and (A, B) non-extendible, and let A = A 1 ∪A 0 be a quasi-periodic decomposition with A 1 non-empty and periodic with maximal period H .…”

“…Thus in view of (22) applied with i = 1, it follows that ρ 2 = 0. Hence, since A is aperiodic (else A + B is periodic), it follows that |H 2 | = 1, whence (23) and (22)…”

Large sets with small doubling modulo p are well covered by an arithmetic progression

“…They are implied by Vosper's theorem [19] (m = −1), by a result of Hamidoune and Rødseth [10] (m = 0) and by a result of Hamidoune and the present authors [11] (m = 1). In the present paper we shall prove conjecture 1.2 for all values of m up to |S|, where is a fixed absolute constant.…”

“…Our strategy is to use an auxiliary set A that minimizes the difference |S + A| − |S| among all sets such that |A| m + 3 and |S + A| p − (m + 3). The set A is called an (m + 3)-atom of S and using such sets to derive properties of S is an instance of the isoperimetric (or atomic) method in additive number theory which was introduced by Hamidoune and developed in [6,7,8,9,17,11,12]. The point of introducing the set A is that we shall manage to prove that it is both significantly smaller than S and also has a small sumset 2A.…”