Bounds on the cardinality of restricted sumsets in $\mathbb{Z}_{p}$

Gabriel Bengochea; Bernardo Llano

Bounds on the cardinality of restricted sumsets in $\mathbb{Z}_{p}$

Combinatorics 2018-03-28 v1

Authors: Gabriel Bengochea , Bernardo Llano

Abstract

In this paper we present a procedure which allows to transform a subset $A$ of $\mathbb{Z}_{p}$ into a set $A'$ such that $|2\hspace{0.15cm}\widehat{} A'|\leq|2\hspace{0.15cm}\widehat{} A |$ , where $2\hspace{0.15cm}\widehat{} A$ is defined to be the set $\left\{a+b:a\neq b,\;a,b\in A\right\}$ . From this result, we get some lower bounds for $|2\hspace{0.15cm}\widehat{} A|$ . Finally, we give some remarks related to the problem for which sets $A\subset \mathbb{Z}_{p}$ we have the equality $|2\hspace{0.15cm}\widehat{} A|=2|A|-1$ .

Keywords

restricted sumsets cardinality bounds Z_p additive combinatorics subset transformation

Cite

@article{arxiv.1803.09400,
  title  = {Bounds on the cardinality of restricted sumsets in $\mathbb{Z}_{p}$},
  author = {Gabriel Bengochea and Bernardo Llano},
  journal= {arXiv preprint arXiv:1803.09400},
  year   = {2018}
}

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16 pages

Bounds on the cardinality of restricted sumsets in $\mathbb{Z}_{p}$

Abstract

Keywords

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