Minimun overlap problem on finite groups

Abstract

Let A, B be disjoint sets such that Acup B= [1,2n]subset {mathbb {Z}} and vert Avert = vert Bvert = n. Let us call m (A, B)=max _{t in {mathbb {Z}}}vert (t+B)cap A vert and consider M(n):=min limits _{(A,B)} m(A,B)(over all partitions with A cup B=left[ 1,2nright] ). There are well-known upper and lower bounds of M(n). In this paper we studied a variation of this problem, i.e. we considered a finite abelian group G with vert G vert =k, we define M(G) which is analogous to M(n) and we obtained upper and lower bounds for M(G).