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**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: **sum**_{c in G} |m(x,E,E)+m(x,E',E')-m(x,E,E')-m(x,E',E)|^{2} =

**sum**_{c in G} |m(x,E,-E)+m(x,E',-E')-m(x,E,-E')-m(x,E',-E)|^{2} (i) (Cohesion equation)

**max**_{E\subset G}**max**_{x 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

**max**_{E\subset G}**max**_{x 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

11P99 Additive number theory

20P05 Probability methods in group theory

11B05 Topology etc. of sets of numbers

**Keywords: ** finite Abelian group; sum set; Cohesin equation

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