Concept

Restricted sumset

Summary
In additive number theory and combinatorics, a restricted sumset has the form :S={a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n \ \mathrm{and}\ P(a_1,\ldots,a_n)\not=0}, where A_1,\ldots,A_n are finite nonempty subsets of a field F and P(x_1,\ldots,x_n) is a polynomial over F. If P is a constant non-zero function, for example P(x_1,\ldots,x_n)=1 for any x_1,\ldots,x_n, then S is the usual sumset A_1+\cdots+A_n which is denoted by nA if A_1=\cdots=A_n=A. When :P(x_1,\ldots,x_n) = \prod_{1 \le i < j \le n} (x_j-x_i), S is written as A_1\dotplus\cdots\dotplus A_n which is denoted by n^{\wedge} A if A_1=\cdots=A_n=A. Note that |S| > 0 if and only if there exist a_1\in A_1,\ldots,a_n\in A_n with P(a_1,\ldots,a_n)\not=0. Cauchy–Davenport theorem The Cauchy–Daven
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