## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  464.20034
Autor:  Erdös, Paul; Smith, B.
Title:  Finite Abelian group cohesion. (In English)
Source:  Isr. J. Math. 39, 177-185 (1981).
Review:  Let G be a finite Abelian group with \#G = p. For A,b\subset G let m(x,A,B) = \#{(a,b): a+b = x,a in A,b in B}. For E\subset G let E' denote its complement. The authors prove the following results:

sumc in G |m(x,E,E)+m(x,E',E')-m(x,E,E')-m(x,E',E)|2 =
sumc in G |m(x,E,-E)+m(x,E',-E')-m(x,E,-E')-m(x,E',-E)|2    (i) (Cohesion equation)

maxE\subset Gmaxx in G|m(x,E,E)+m(x,E',E')-2m(x,E,E')| \geq p ½    (ii)

If \lambda > ½ and G contains no element of order 2, then

maxE\subset Gmaxx in G|m(x,E,E)+m(x,E',E')-2m(x,E,E')| \geq K.p\lambda    (iii)

Here K depends only on \lambda.
Reviewer:  St.Porubský
Classif.:  * 20K01 Finite abelian groups
20D60 Arithmetic and combinatorial problems on finite groups