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{\bf Vsevolod F. Lev}
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{\bf \raggedright Restricted set addition in groups, II.  \break
A generalization of the Erd\H os-Heilbronn conjecture}
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In 1980, Erd\H os and Heilbronn posed the problem of estimating (from below)
the number of sums $a+b$ where $a\in A$ and $b\in B$ range over given sets
$A,B\subseteq{\Bbb Z}/p{\Bbb Z}$ of residues modulo a prime $p$, so that $a\neq b$. A solution
was given in 1994 by Dias da Silva and Hamidoune. In 1995, Alon, Nathanson
and Ruzsa developed a polynomial method that allows one to handle restrictions
of the type $f(a,b)\neq 0$, where $f$ is a polynomial in two variables over
${\Bbb Z}/p{\Bbb Z}$.

In this paper we consider restricting conditions of general type and
investigate groups, distinct from ${\Bbb Z}/p{\Bbb Z}$. In particular, for $A,B\subseteq{\Bbb Z}/p{\Bbb Z}$ and
${\cal R}\subseteq A\times B$ of given cardinalities we give a sharp estimate for the
number of distinct sums $a+b$ with $(a,b)\notin\ {\cal R}$, and we obtain a partial
generalization of this estimate for arbitrary Abelian groups.


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