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Some inverse results of sumsets

Min Tang|Yun Xing
대한수학회 (The Korean Mathematical Society)|2021.03|대한수학회보 (Bulletin of the KMS)|제58권 제2호|pp. 305-313|DOI 10.4134/BKMS.b200054

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키워드

inverse problemSumsetsarithmetic progression
주제분류수학

초록

Let $h\geq 2$ and $A=\{a_0,a_1,\ldots,a_{k-1}\}$ be a finite set of integers. It is well-known that $\left|hA\right|=hk-h+1$ if and only if $A$ is a $k$-term arithmetic progression. In this paper, we give some nontrivial inverse results of the sets $A$ with some extremal the cardinalities of $hA$.

Abstract

Let $h\geq 2$ and $A=\{a_0,a_1,\ldots,a_{k-1}\}$ be a finite set of integers. It is well-known that $\left|hA\right|=hk-h+1$ if and only if $A$ is a $k$-term arithmetic progression. In this paper, we give some nontrivial inverse results of the sets $A$ with some extremal the cardinalities of $hA$.

참고문헌 (13)

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  3. G. A. Fre˘ıman, "Inverse problems of additive number theory. VI. On the addition of finite sets. III", Izv. VysS. Uˇcebn. Zaved. Matematika, 3(28), pp.151~157 (1962)
  4. G. A. Fre˘ıman, "Combinatorial number theory and additive group theory", pp.233~239 (2009), Birkh¨auser Verlag
  5. Gregory A. Freiman, "Inverse Additive Number Theory. XI. Long arithmetic progressions in sets with small sumsets", Acta Arithmetica, 137(4), pp.325~331 (2009), Institute of Mathematics, Polish Academy of Sciences[DOI]
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  9. Melvin B. Nathanson, "Sums of Finite Sets of Integers", The American Mathematical Monthly, 79(9), pp.1010~1012 (1972), JSTOR[DOI]
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  13. Quan-Hui Yang, "On the cardinality of general h-fold sumsets", European Journal of Combinatorics, 47, pp.103~114 (2015), Elsevier BV[DOI]

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