Combinatorial Number Theory and Additive Group Theory by Alfred Geroldinger

By Alfred Geroldinger

Additive combinatorics is a comparatively contemporary time period coined to understand the advancements of the extra classical additive quantity conception, almost always focussed on difficulties relating to the addition of integers. a few classical difficulties just like the Waring challenge at the sum of k-th powers or the Goldbach conjecture are real examples of the unique questions addressed within the quarter. one of many beneficial properties of up to date additive combinatorics is the interaction of a good number of mathematical innovations, together with combinatorics, harmonic research, convex geometry, graph conception, chance concept, algebraic geometry or ergodic idea. This booklet gathers the contributions of some of the prime researchers within the quarter and is split into 3 elements. the 2 first elements correspond to the cloth of the most classes introduced, Additive combinatorics and non-unique factorizations, via Alfred Geroldinger, and Sumsets and constitution, by way of Imre Z. Ruzsa. The 3rd half collects the notes of many of the seminars which followed the most courses, and which hide a fairly large a part of the equipment, innovations and difficulties of up to date additive combinatorics.

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The question arose, which is still wide open, whether the same is true for the system of sets of lengths. Clearly, if H and H are reduced Krull monoids with isomorphic class groups G, G and primes in all classes, then L(H) = L(G) = L(H ), but H and H need not be isomorphic. 2]) that L(C3 ) = {y + 2k + [0, k] | y, k ∈ N0 } = L(C2 ⊕ C2 ) . Note that D(C3 ) = D(C2 ⊕ C2 ) = 3, and C1 , C2 , C2 ⊕ C2 and C3 are the only finite abelian groups G with D(G ) ≤ 3. So the best we can hope for is a positive answer to the following question: Given two finite abelian groups G and G with D(G) ≥ 4 such that L(G) = L(G ), does it follow that G ∼ =G?

Wt ∈ A(G). By the induction hypothesis there is a d-chain of factorizations y0 , . . , yk concatenating U1 ·. ·Ur−1 and V1 W1 ·. ·Wt , and there is a d-chain of factorizations z0 , . . , zl concatenating W1 · . . · Wt Ur and V2 · . . · Vs . Then z = y0 Ur , . . , yk Ur = z0 V1 , . . , zl V1 = z is a d-chain concatenating z and z . Case 2: |Vi | > d for all i ∈ [1, s]. By assumption there is a factorization V1 V2 = W1 · . . · Wk , where k ∈ [2, d] and |W1 | ≤ d. Then the factorization z = W1 · .

Gl ∈ A(G) with |U | = l = D(G), then D(G) 2 . If U = l (−U )k U k = (−gν )gν k ν=1 shows that in both inequalities we actually have equality. 4. Let j, l ∈ N0 such that lD(G) + j ≥ 1. Then 2. and 3. imply that 2j = ρ(G)−1 lD(G) + j ≤ λlD(G)+j (G) ≤ lD(G) + j . D(G) If j = 0, it follows that λlD(G) (G) = 2l. 2. 1. For j ∈ N≥2 , the following statements are equivalent : (a) There exists some L ∈ L(G) with {2, j} ⊂ L. (b) j ≤ D(G). 2. Let |G| ≥ 3 and A ∈ B(G). Then {2, D(G)} ⊂ L(A) if and only if A = U (−U ) for some U ∈ A(G) with |U | = D(G).

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