Bhanja, Jagannath A note on sumsets and restricted sumsets. (English) Zbl 1480.11125 J. Integer Seq. 24, No. 4, Article 21.4.2, 9 p. (2021). This paper is a contribution to additive combinatorics, and more precisely to sumset theory. For a given set \(A\) on an ambient abelian group \((G,+)\), let us write \(hA=A+\stackrel{h}{\dots}+A\), where \(A+A=\{a+a':a,\,a'\in A\}\). Similarly, we define \(h\hat{\,}A\) as the sum of elements in \(A\) were elements are not allowed to be equal.For a given set of integers \(H\), we write \[HA=\cup_{h\in H}hA\] and similarly we can define \(H\hat{\,}A\). The main general results of this paper deals with the following question: once we know the size of \(A\) and \(H\), can we give lower bounds for the size of \(HA\) and \(H\hat{\,}A\)? and, once a couple of sets \(H\) and \(A\) reach such lower bound, can we say ‘how’ is \(A\)?In the case of \(HA\) Theorem 3 solves the first question when \(A\) contains only positive integers, and Theorem 5 gives an answer to the inverse result. Theorem 6 and 9 answer the questions in the case of \(H\hat{\,}A\). Reviewer: Juanjo Rué Perna (Barcelona) Cited in 2 Documents MSC: 11P70 Inverse problems of additive number theory, including sumsets 11B75 Other combinatorial number theory 11B13 Additive bases, including sumsets Keywords:sumset; restricted sumset PDFBibTeX XMLCite \textit{J. Bhanja}, J. Integer Seq. 24, No. 4, Article 21.4.2, 9 p. (2021; Zbl 1480.11125) Full Text: arXiv Link References: [1] B. Bajnok,Additive Combinatorics: A Menu of Research Problems, Discrete Mathematics and its Applications, CRC Press, 2018. · Zbl 1415.11001 [2] ´E. Balandraud, Addition theorems inFpvia the polynomial method, preprint, 2017. Available athttps://arxiv.org/abs/1702.06419. [3] J. Bhanja and R. K. Pandey, Inverse problems for certain subsequence sums in integers, Discrete Math.343(2020), 112148. · Zbl 1458.11016 [4] M. B. Nathanson, Inverse theorems for subset sums,Trans. Amer. Math. Soc.347(1995), 1409-1418. · Zbl 0835.11006 [5] M. B. Nathanson,Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Springer, 1996. · Zbl 0859.11003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.