A Survey of Problems and Results on Restricted Sumsets
نویسنده
چکیده
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. 1. Erdős-Heilbronn conjecture and the polynomial method Let A = {a1, . . . , ak} and B = {b1, . . . , bl} be two finite subsets of Z with a1 < · · · < ak and b1 < · · · < bl. Observe that a1 + b1 < a2 + b1 < · · · < ak + b1 < ak + b2 < · · · < ak + bl, whence 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 {Ai}i=1 of sets, a sequence {ai}i=1 is called a system of distinct representatives of {Ai}i=1 if a1 ∈ A1, . . . , an ∈ An and a1, . . . , an are distinct. A fundamental theorem of P. Hall [17] states that {Ai}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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