An inverse theorem for the restricted set addition in Abelian groups

Károlyi, György

doi:10.1016/j.jalgebra.2005.04.021

Cited by 23 publications

(17 citation statements)

References 25 publications

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“…If m = k − 1, then As far as the inverse problem modulo a prime is concerned, in [7] the inverse theorem of the Erdős-Heilbronn conjecture is proved. The proof however works only when adding two copies of A and, to the best of the author's knowledge, an inverse theorem for hˆA, h > 2, does not exist yet.…”

Section: Inverse Problemmentioning

confidence: 99%

A generalization of sumsets modulo a prime

Monopoli

2015

Journal of Number Theory

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Let A be a set in an abelian group G. For integers h, r ≥ 1 the generalized hfold sumset, denoted by h (r) A, is the set of sums of h elements of A, where each element appears in the sum at most r times. If G = Z lower bounds for |h (r) A| are known, as well as the structure of the sets of integers for which |h (r) A| is minimal. In this paper we generalize this result by giving a lower bound for |h (r) A| when G = Z/pZ for a prime p, and show new proofs for the direct and inverse problems in Z.

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Section: Inverse Problemmentioning

confidence: 99%

A generalization of sumsets modulo a prime

Monopoli

2015

Journal of Number Theory

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“…G. Károlyi ([19], [20]) extended the Erdős-Heilbronn conjecture to general abelian groups. Theorem 5.2.…”

Section: Working With General Abelian Groupsmentioning

confidence: 99%

“…(ii) (G. Károlyi [20]) When |A| 5 and p(G) > 2|A| − 3, the equality |2 ∧ A| = 2|A| − 3 holds if and only if A is an arithmetic progression.…”

Section: Working With General Abelian Groupsmentioning

confidence: 99%

A Survey of Problems and Results on Restricted Sumsets

Sun¹

2007

Number Theory

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Additive number theory is currently an active field related to combinatorics. In this paper we give a survey of problems and results concerning lower bounds for cardinalities of various restricted sumsets with elements in a field or an abelian group. Erdős-Heilbronn conjecture and the polynomial methodLet A = {a 1 , . . . , a k } and B = {b 1 , . . . , b l } be two finite subsets of Z with a 1 < · · · < a k andwhence we see that the sumset A + B = {a + b: a ∈ A and b ∈ B} contains at least k + l − 1 elements. In particular, |2A| 2|A| − 1, where |A| denotes the cardinality of A, and 2A stands for A + A.The following fundamental theorem was first proved by A. Cauchy [9] in 1813 and then rediscovered by H. Davenport [11] in 1935.Cauchy-Davenport Theorem. Let A and B be non-empty subsets of the field Z/pZ where p is a prime. Then |A + B| min{p, |A| + |B| − 1}.(1.1)For lots of important results on sumsets over Z, the reader is referred to the recent book [38] by T. Tao and V. H. Vu. In this paper we mainly focus our attention on restricted sumsets with elements in a field or an abelian group.In combinatorics, for a finite sequence {A i } n i=1 of sets, a sequence {a i } n i=1 is called a system of distinct representatives of {A i } n i=1 if a 1 ∈ A 1 , . . . , a n ∈ A n and a 1 , . . . , a n are distinct. A fundamental theorem of P. Hall [17] states that {A i } n i=1 has a system of distinct Supported by the National Science Fund (No. 10425103) for Distinguished Young Scholars in China.

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“…This is an extension of the Cauchy-Davenport theorem for finite groups. In [7,8], Károlyi established the following generalizaton of the the Erdős-Heilbronn problem:where A is a non-empty subset of the finite abelian group G. Subsequently, Balister and Wheeler [3] removed the restriction that G is abelian. In fact, they showed that |A ∔ B| ≥ min{p(G), |A| + |B| − 3}, for any non-empty subsets A, B of a finite group G.In this paper, we shall consider the extension of (1.1) for finite nilpotent groups.Theorem 1.1.…”

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confidence: 99%