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A note on sumsets and restricted sumsets. (English) Zbl 1480.11125

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 \(H\hat{\,}A\).

MSC:

11P70 Inverse problems of additive number theory, including sumsets
11B75 Other combinatorial number theory
11B13 Additive bases, including sumsets
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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
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