32 research outputs found
Direct and inverse problems for restricted signed sumsets in integers
Let be a nonempty finite subset of an additive abelian group . For a positive integer , we let
be the -fold restricted signed sumset of . The direct problem for the restricted signed sumset is to find the minimum number of elements in in terms of , where is the cardinality of . The {\it inverse problem} for the restricted signed sumset is to determine the structure of the finite set for which the minimum value of is achieved. In this article, we solve some cases of both direct and inverse problems for in the group of integers. In this connection, we also mention some conjectures in the remaining cases
Direct and inverse problems for restricted signed sumsets in integers
Let be a nonempty finite subset of an additive abelian group . For a positive integer , we let
be the -fold restricted signed sumset of . The direct problem for the restricted signed sumset is to find the minimum number of elements in in terms of , where is the cardinality of . The {\it inverse problem} for the restricted signed sumset is to determine the structure of the finite set for which the minimum value of is achieved. In this article, we solve some cases of both direct and inverse problems for in the group of integers. In this connection, we also mention some conjectures in the remaining cases
Additive Combinatorics: A Menu of Research Problems
This text contains over three hundred specific open questions on various
topics in additive combinatorics, each placed in context by reviewing all
relevant results. While the primary purpose is to provide an ample supply of
problems for student research, it is hopefully also useful for a wider
audience. It is the author's intention to keep the material current, thus all
feedback and updates are greatly appreciated.Comment: This August 2017 version incorporates feedback and updates from
several colleague
Sumset Valuations of Graphs and Their Applications
International audienc
Sets with few distinct distances do not have heavy lines
Let be a set of points in the plane that determines at most
distinct distances. We show that no line can contain more than points of . We also show a similar result for rectangular
distances, equivalent to distances in the Minkowski plane, where the distance
between a pair of points is the area of the axis-parallel rectangle that they
span